90-Minute Classes. Module 21. DAY 1. DAY 2. DAY 3. Lesson 21.1. Lesson 21.3. Module Review and. Assessment Readiness. Lesson 21.2. Lesson 21.3 ...... ...

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Volume CONTENTS COMMON CORE

G-GMD.A.1 G-GMD.A.1 G-GMD.A.1 G-GMD.A.2 G-GMD.A.3

1117A

Unit 9

MODULE 21

Volume Formulas

Lesson 21.1 Lesson 21.2 Lesson 21.3 Lesson 21.4 Lesson 21.5

Volume of Prisms and Cylinders Volume of Pyramids . . . . . . . Volume of Cones . . . . . . . . . . Volume of Spheres . . . . . . . . Scale Factor . . . . . . . . . . . . .

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UNIT 9

Unit Pacing Guide 45-Minute Classes Module 21 DAY 1

DAY 2

DAY 3

DAY 4

DAY 5

Lesson 21.1

Lesson 21.2

Lesson 21.3

Lesson 21.4

Lesson 21.5

DAY 6

DAY 7

Module Review and Assessment Readiness

Unit Review and Assessment Readiness

90-Minute Classes Module 21 DAY 1

DAY 2

DAY 3

Lesson 21.1

Lesson 21.3

Module Review and Assessment Readiness

Lesson 21.2 Lesson 21.3

Lesson 21.4 Lesson 21.5

Unit Review and Assessment Readiness

Unit 9

1117B

Program Resources PLAN

ENGAGE AND EXPLORE

HMH Teacher App Access a full suite of teacher resources online and offline on a variety of devices. Plan present, and manage classes, assignments, and activities.

Real-World Videos Engage students with interesting and relevant applications of the mathematical content of each module.

Explore Activities Students interactively explore new concepts using a variety of tools and approaches.

ePlanner Easily plan your classes, create and view assignments, and access all program resources with your online, customizable planning tool.

Professional Development Videos Authors Juli Dixon and Matt Larson model successful teaching practices and strategies in actual classroom settings. QR Codes Scan with your smart phone to jump directly from your print book to online videos and other resources. DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A

Teacher’s Edition Support students with point-of-use Questioning Strategies, teaching tips, resources for differentiated instruction, additional activities, and more. DONOT NOTEDIT--Changes EDIT--Changesmust mustbe bemade madethrough through"File "Fileinfo" info" DO CorrectionKey=NL-A;CA-A CorrectionKey=NL-A;CA-A

Name Name

Isosceles and Equilateral Triangles

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Proving the Isosceles Triangle Theorem and Its Converse

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PREVIEW: LESSON PERFORMANCE TASK

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Language Objective

Essential Question: What are the special relationships among angles and sides in isosceles and equilateral triangles?

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An isosceles triangle is a triangle with at least two congruent sides.

Students have the option of completing the isosceles triangle activity either in the book or online.

Resource Locker

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LESSON

Name

Base Base angles

PROFESSIONAL DEVELOPMENT

TEACH

ASSESSMENT AND INTERVENTION

Math On the Spot video tutorials, featuring program authors Dr. Edward Burger and Martha Sandoval-Martinez, accompany every example in the textbook and give students step-by-step instructions and explanations of key math concepts.

Interactive Teacher Edition Customize and present course materials with collaborative activities and integrated formative assessment.

C1

Lesson 19.2 Precision and Accuracy

Evaluate

1

Lesson XX.X ComparingLesson Linear, Exponential, and Quadratic Models 19.2 Precision and Accuracy

teacher Support

1

EXPLAIN Concept 1

Explain

The Personal Math Trainer provides online practice, homework, assessments, and intervention. Monitor student progress through reports and alerts. Create and customize assignments aligned to specific lessons or Common Core standards. • Practice – With dynamic items and assignments, students get unlimited practice on key concepts supported by guided examples, step-by-step solutions, and video tutorials. • Assessments – Choose from course assignments or customize your own based on course content, Common Core standards, difficulty levels, and more. • Homework – Students can complete online homework with a wide variety of problem types, including the ability to enter expressions, equations, and graphs. Let the system automatically grade homework, so you can focus where your students need help the most! • Intervention – Let the Personal Math Trainer automatically prescribe a targeted, personalized intervention path for your students. 2

3

4

Question 3 of 17

Concept 2

Determining Precision

ComPLEtINg thE SquArE wIth EXPrESSIoNS Avoid Common Errors Some students may not pay attention to whether b is positive or negative, since c is positive regardless of the sign of b. Have student change the sign of b in some problems and compare the factored forms of both expressions. questioning Strategies In a perfect square trinomial, is the last term always positive? Explain. es, a perfect square trinomial can be either (a + b)2 or (a – b)2 which can be factored as (a + b)2 = a 2 + 2ab = b 2 and (a – b)2 = a 2 + 2ab = b 2. In both cases the last term is positive. reflect 3. The sign of b has no effect on the sign of c because c = ( b __ 2 ) 2 and a nonzero number squared is always positive. Thus, c is always positive. c = ( b __ 2 ) 2 and a nonzero number c = ( b __ 2 ) 2 and a nonzero number

5

6

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View Step by Step

8

9

10

11 - 17

Video Tutor

Personal Math Trainer

Textbook

X2 Animated Math

Solve the quadratic equation by factoring. 7x + 44x = 7x − 10

As you have seen, measurements are given to a certain precision. Therefore,

x=

the value reported does not necessarily represent the actual value of the measurement. For example, a measurement of 5 centimeters, which is

,

Check

given to the nearest whole unit, can actually range from 0.5 units below the reported value, 4.5 centimeters, up to, but not including, 0.5 units above it, 5.5 centimeters. The actual length, l, is within a range of possible values:

Save & Close

centimeters. Similarly, a length given to the nearest tenth can actually range from 0.05 units below the reported value up to, but not including, 0.05 units above it. So a length reported as 4.5 cm could actually be as low as 4.45 cm or as high as nearly 4.55 cm.

?

!

Turn It In

Elaborate

Look Back

Focus on Higher Order Thinking Raise the bar with homework and practice that incorporates higher-order thinking and mathematical practices in every lesson.

Differentiated Instruction Resources Support all learners with Differentiated Instruction Resources, including • Leveled Practice and Problem Solving • Reading Strategies • Success for English Learners • Challenge Calculate the minimum and maximum possible areas. Round your answer to

Assessment Readiness

the nearest square centimeters.

The width and length of a rectangle are 8 cm and 19.5 cm, respectively.

Prepare students for success on high stakes tests for Integrated Mathematics 2 with practice at every module and unit

Find the range of values for the actual length and width of the rectangle.

Minimum width =

7.5

cm and maximum width <

8.5 cm

My answer

Assessment Resources

Find the range of values for the actual length and width of the rectangle.

Minimum length =

19.45

cm and maximum length < 19.55

Name ________________________________________ Date __________________ Class __________________ LESSON

1-1

cm

Name ____________ __________________ __________ Date __________________ LESSON Class ____________ ______

Precision and Significant Digits

6-1

Success for English Learners

Linear Functions

Reteach

The graph of a linear The precision of a measurement is determined bythe therange smallest unit or Find of values for the actual length and width of the rectangle. function is a straig ht line. fraction of a unit used. Ax + By + C = 0 is the standard form for the equat ion of a linear functi • A, B, and C are on. Problem 1 Minimum Area = Minimum width × Minimum length real numbers. A and B are not both zero. • The variables x and y Choose the more precise measurement. = 7.5 cm × 19.45 cm have exponents of 1 are not multiplied together are not in denom 42.3 g is to the 42.27 g is to the inators, exponents or radical signs. nearest tenth. nearest Examples These are NOT hundredth. linear functions: 2+4=6 no variable x2 = 9 exponent on x ≥ 1 xy = 8 x and y multiplied 42.3 g or 42.27 g together 6 =3 Because a hundredth of a gram is smaller than a tenth of a gram, 42.27 g x in denominator x is more precise. 2y = 8 y in exponent Problem 2 In the above exercise, the location of the uncertainty in the linear y = 5 y in a square root measurements results in different amounts of uncertainty in the calculated Choose the more precise measurement: 36 inches or 3 feet. measurement. Explain how to fix this problem. Tell whether each function is linear or not. 1. 14 = 2 x 2. 3xy = 27 3. 14 = 28 4. 6x 2 = 12 x ____________

Reflect

____

________________

_______________

The graph of y = C is always a horiz ontal line. The graph always a vertical line. of x = C

_______________

is

Unit 9

Send to Notebook

_________________________________________________________________________________________

2. An object is weighed on three different scales. The results are shown Explore in the table. Which scale is the most precise? Explain your answer. Measurement

____________________________________________________________

• Tier 1, Tier 2, and Tier 3 Resources

Examples

1. When deciding which measurement is more precise, what should you Formula consider?

Scale

Tailor assessments and response to intervention to meet the needs of all your classes and students, including • Leveled Module Quizzes • Leveled Unit Tests • Unit Performance Tasks • Placement, Diagnostic, and Quarterly Benchmark Tests

Your Turn

y=1 T

x=2

y = −3

x=3

1117D

Math Background Volume

COMMON CORE

G-GMD.A.2

LESSONS 21.1 to 21.4 The formula for the volume of a rectangular prism (V = Bh) is the starting point for developing the volume formulas for other three-dimensional figures. Another important ingredient in developing volume formulas is Cavalieri’s Principle. This principle says that if two three-dimensional figures have the same height and the same cross-sectional area at every level, then they have the same volume. To illustrate the use of Cavalieri’s Principle, consider the following informal argument for the volume formula for a cylinder with height h and base area B. Construct a rectangular prism of height h so that each rectangular cross-section has area B. As shown in the following figure, the prism and cylinder can be positioned so that the area of any cross-section of the cylinder created by a plane parallel to the base has the same area as the corresponding cross-section of the prism. By Cavalieri’s Principle, the volume of the cylinder is equal to that of the prism. That is, V = Bh. B

B h

Note that the above argument works for all cylinders, oblique or right, and regardless of the shape of the base. In the case of a cylinder with a circular base, the volume formula may be written as V = πr 2h. For pyramids, cones, and spheres, the situation is somewhat more complex. Students can fill models of solids with sand or water to gain an intuitive sense of how the volumes of pyramids and cones are related to the volumes of prisms and cylinders, respectively. However, rigorous justifications of the volume formulas for pyramids, cones, and spheres all rely on clever applications of Cavalieri’s Principle.

1117E

Unit 9

Students can use a stack of pennies to understand Cavalieri’s principle. Have students arrange the pennies to form a right cylinder and ask them to estimate the volume. Then, have students push the stack to form an oblique cylinder. Students should see that the volume of the stack does not change. This is supported by Cavalieri’s principle because the cross-sectional area at each level (that is, the area of the face of a penny) is unchanged when the stack is pushed to formed the oblique cylinder. Students may have worked primarily with two-dimensional figures. Here the focus shifts to three-dimensional figures. Students have probably already explored three-dimensional figures in earlier grades, especially in the context of volume problems. Students will learn and use the essential volume formulas. The formula for the volume, V, of a prism and the volume, V, of a right or oblique cylinder is the same, V = Bh, where B is the area of the base and h is the height. Review the definition of pyramid and the associated vocabulary (lateral face, vertex, base). To sketch a pyramid, start by drawing a polygonal base and plotting a point for the vertex. Then draw straight lines from the vertex to each vertex of the polygonal base. Use dashed lines for edges that are hidden when the pyramid is viewed from the front. A key postulate about pyramids states that pyramids with equal base areas and equal heights have equal volumes. A wedge pyramid is a pyramid in which the base is a triangle and a perpendicular segment from the pyramid’s vertex to the base intersects the base at a vertex of the triangle. Any pyramid can be divided into wedge pyramids.

PROFESSIONAL DEVELOPMENT

The following hands-on activity can help students develop the formula for the volume of a pyramid from an inductivereasoning perspective. Have students make nets for a square-based pyramid and a square-based prism that has the same height as the pyramid. Then, have students cut out, fold, and tape the nets to form the three-dimensional figures. Students can model the volume of the pyramid by filling it with uncooked rice, sand, or another granular material. Ask students to pour the rice from the pyramid into the prism as many times as necessary to see how the volumes of the figures are related. Students will discover that it takes three batches of rice from the pyramid to fill the prism. That is, the volume of the pyramid is one-third the volume of the associated prism. One approach to finding a formula for the volume of a cone is similar to a method for finding a formula for the circumference of a circle. This can be done by using inscribed regular polygons and an informal limit argument to show that the circumference, C, of a circle with radius r is given by C = 2πr. You can inscribe a sequence of pyramids in a given cone and use similar reasoning to show that the volume, V, of the cone is given by V = __13 Bh, where B is the base area and h is the cone’s height.

One way to develop a formula for the volume of a sphere offers a surprising application of Cavalieri’s principle. It is surprising because the argument is based on showing that two seemingly unrelated solids have the same volume. The solids—a hemisphere and a cylinder from which a cone has been removed—are shown to have the same crosssectional area at every level and therefore must have the same volume. The formula for the volume, V, of a sphere with radius r is V = __43 πr3. A bit of algebra shows that the volume of a sphere is equal to __23 the volume of its circumscribed cylinder. This result, which has been known since ancient times, was of such importance to the Greek mathematician Archimedes that he requested that a drawing of a sphere and cylinder be placed on his tomb.

Scale Factor

COMMON CORE

G-GMD.A.3

LESSON 21.5 When all dimensions of a figure are multiplied by a nonzero constant k, the perimeter or circumference changes by a factor of k and the area changes by a factor of k 2. This principle can be proved for various categories of figures by using established formulas. Consider the case of a triangle, ▵ABC, with sides of length a, b, and c. When all dimensions are multiplied by k (k ≠ 0), the resulting triangle has sides of length ka, kb, and kc. The perimeter of the new triangle is therefore ka + kb + kc or k(a + b + c), which is k times the perimeter of ▵ABC. The three-dimensional analogue of this principle says that, when all the dimensions of a three-dimensional figure are multiplied by a nonzero constant k, the surface area changes by a factor of k 2 and the volume changes by a factor of k 3.

Unit 9

1117F

UNIT

9

UNIT 9

Volume

MODULE

Volume

MATH IN CAREERS Unit Activity Preview

21

Volume Formulas

After completing this unit, students will complete a Math in Careers task by investigating the volume of a piece of jewelry. Critical skills include modeling real-world situations and applying volume formulas. For more information about careers in mathematics as well as various mathematics appreciation topics, visit The American Mathematical Society at http://www.ams.org.

MATH IN CAREERS

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Hero Images/Corbis

Jewelry Maker A jewelry maker designs and creates jewelry. Jewelry makers often employ geometric designs and shapes in their work, and so they need a good understanding of geometry. For example, they must calculate volume and surface area to determine the amount of materials needed. They can also use computer designing programs to help them with their design specifications. Jewelry makers often need to calculate costs of materials and labor to determine production costs for their designs. If you are interested in a career as a jewelry maker, you should study these mathematical subjects: • Algebra • Geometry • Business Math Research other careers that require knowing the geometry of threedimensional objects. Check out the career activity at the end of the unit to find out how jewelry makers use math. Unit 9

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TRACKING YOUR LEARNING PROGRESSION

IN2_MNLESE389847_U9UO 1117

4/19/14 9:10 AM

Before

In this Unit

After

Students understand: • central and inscribed angles • chords, secants, tangent lines, and arcs • inscribed quadrilaterals • segment lengths in circles • formulas for circumference, area, and equation of a circle • arc lengths, concentric circles, radian measure • area of a sector

Students will learn about: • formulas for the volume of a prism, cylinder, pyramid, cone, and sphere • scale factor

Students will study: • probability • permutations and combinations • conditional probability • independent and dependent events • making and analyzing decisions using probability

1117

Unit 9

Reading Start -Up

Reading Start Up

Vocabulary Review Words

Visualize Vocabulary Use the ✔ words and draw examples to complete the chart. Object

Have students complete the activities on this page by working alone or with others.

area (área) composite figure (figura compuesta) ✔ cone (cono)

✔ cylinder (cilindro) ✔ pyramid (pirámide) ✔ sphere (esfera) volume (volume)

Example

VISUALIZE VOCABULARY The example chart helps students review vocabulary associated with three-dimensional figures. If time allows, discuss the characteristics that helped students identify each figure.

Preview Words

cone

apothem (apotema) oblique cylinder (cilindro oblicuo) oblique prism (prisma oblicuo) regular pyramid (pirámide regular) right cone (cono recto) right cylinder (cilindro recto) right prism (prisma recto)

cylinder

UNDERSTAND VOCABULARY Use the following explanations to help students learn the preview words. A regular pyramid has a base that is a regular polygon and lateral faces that are congruent isosceles triangles.

pyramid

Understand Vocabulary Complete the sentences using the preview words. 1. 2.

A cone whose axis is perpendicular to its base is called a(n) right cone . A prism that has at least one nonrectangular lateral face is called a(n) oblique prism .

Active Reading

© Houghton Mifflin Harcourt Publishing Company

sphere

A cylinder whose axis is perpendicular to its bases is a right cylinder. A cylinder in which this is not the case is an oblique cylinder.

ACTIVE READING Students can use these reading and note-taking strategies to help them organize and understand the new concepts and vocabulary. Encourage students to make connections among the figures and their properties by using descriptive vocabulary when constructing their vocabulary pyramids. Remind students to keep asking questions about any vocabulary that they find confusing or unclear.

Pyramid Create a Pyramid and organize the adjectives used to describe different objects—right, regular, oblique—on each of its faces. When listening to descriptions of objects, look for these words and associate them with the object that follows.

Unit 9

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ADDITIONAL RESOURCES IN2_MNLESE389847_U9UO 1118

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Differentiated Instruction • Reading Strategies

Unit 9

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MODULE

21

Volume Formulas

Volume Formulas ESSENTIAL QUESTION: Answer: For one example, volume formulas are useful when you want to find how much liquid something can hold, such as a cup or a swimming pool.

Essential Question: How can you use volume

formulas to solve real-world problems?

21 MODULE

LESSON 21.1

Volume of Prisms and Cylinders LESSON 21.2

Volume of Pyramids LESSON 21.3

This version is for Algebra 1 and Geometry only

Volume of Cones

PROFESSIONAL DEVELOPMENT VIDEO

LESSON 21.4

Volume of Spheres

Professional Development Video

LESSON 21.5

Scale Factor

Professional Development my.hrw.com

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Stringer/ Reuters/Corbis

Author Juli Dixon models successful teaching practices in an actual high-school classroom.

REAL WORLD VIDEO Check out how volume formulas can be used to find the volumes of real-world objects, including sinkholes.

MODULE PERFORMANCE TASK PREVIEW

How Big Is That Sinkhole? In 2010, a giant sinkhole opened up in a neighborhood in Guatemala and swallowed up the three-story building that stood above it. In this module, you will choose and apply an appropriate formula to determine the volume of this giant sinkhole.

Module 21

DIGITAL TEACHER EDITION IN2_MNLESE389847_U9M21MO 1119

Access a full suite of teaching resources when and where you need them: • Access content online or offline • Customize lessons to share with your class • Communicate with your students in real-time • View student grades and data instantly to target your instruction where it is needed most

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Module 21

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PERSONAL MATH TRAINER Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, Common Core-aligned practice tests.

4/19/14 8:57 AM

Are YOU Ready?

Are You Ready?

Complete these exercises to review skills you will need for this module.

Area of a Circle Example 1

ASSESS READINESS

Find the area of a circle with radius equal to 5. A = πr 2

Write the equation for the area of a circle of radius r.

A = π(5)

2

Substitute the radius.

A = 25π

Use the assessment on this page to determine if students need strategic or intensive intervention for the module’s prerequisite skills.

• Online Homework • Hints and Help • Extra Practice

Simplify.

Find each area. 1.

A circle with radius 4

16π

2.

A circle with radius 5

25π

3.

A circle with radius 3π

9π 3

4.

2 A circle with radius _ π

π

ASSESSMENT AND INTERVENTION

__4

Volume Properties Example 2

Find the number of cubes that are 1 cm 3 in size that fit into a cube of size 1 m 3. Notice that the base has a length and width of 1 m or 100 cm, so its area is 1 m 2 or 10,000 cm 2. The 1 m 3 cube is 1 m or 100 cm high, so multiply the area of the base by the height to find the volume of 1,000,000 cm 3.

3 2 1

Find the volume. 5.

The volume of a 1 km 3 piece of land in m 3 1,000,000,000 m 3

6.

The volume of a 1 ft 3 box in in. 3 1728 in 3

Personal Math Trainer will automatically create a standards-based, personalized intervention assignment for your students, targeting each student’s individual needs!

Find the area of a rectangular prism with height 4 cm, length 3 cm, and width 5 cm. V = Bh

Write the equation for the volume of a rectangular prism.

V = 60 cm 3

Simplify.

V = (3)(5)(4) The volume of a rectangular prism is the area of the base times the height.

Find each volume. 7.

A rectangular prism with length 3 m, width 4 m, and height 7 m 84 m 3

8.

A rectangular prism with length 2 cm, width 5 cm, and height 12 cm 120 cm 3

Module 21

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Tier 1 Lesson Intervention Worksheets Reteach 21.1 Reteach 21.2 Reteach 21.3 Reteach 21.4 Reteach 21.5

© Houghton Mifflin Harcourt Publishing Company

Volume of Rectangular Prisms Example 3

TIER 1, TIER 2, TIER 3 SKILLS

ADDITIONAL RESOURCES See the table below for a full list of intervention resources available for this module. Response to Intervention Resources also includes: • Tier 2 Skill Pre-Tests for each Module • Tier 2 Skill Post-Tests for each skill

1120

Response to Intervention Tier 2 Strategic Intervention Skills Intervention Worksheets 42 Surface Area 44 Volume

Differentiated Instruction

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Tier 3 Intensive Intervention Worksheets available online Building Block Skills 9, 10, 11, 14, 30, 31, 77, 83, 101, 106

Challenge worksheets Extend the Math Lesson Activities in TE

Module 21 1120

LESSON

21.1

Name

Volume of Prisms and Cylinders

Class

Date

21.1 Volume of Prisms and Cylinders Essential Question: How do the formulas for the volume of a prism and cylinder relate to area formulas that you already know? Resource Locker

Common Core Math Standards The student is expected to: COMMON CORE

Explore

G-GMD.A.1

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Also G-GMD.A.2, G-GMD.A.3, G-MG.A.1, G-MG.A.2

Mathematical Practices COMMON CORE

Developing a Basic Volume Formula

The volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure. This prism is made up of 8 cubes, each with a volume of 1 cubic centimeter, so it has a volume of 8 cubic centimeters. You can use this idea to develop volume formulas.

Volume = 1 cubic unit

In this activity you’ll explore how to develop a volume formula for a right prism and a right cylinder.

MP.4 Modeling

A right prism has lateral edges that are perpendicular to the bases, with faces that are all rectangles.

Language Objective

A right cylinder has bases that are perpendicular to it center axis.

Explain to a partner how to apply the formulas for the volume of a prism and a cylinder. axis

ENGAGE

The formula for the volume of a prism involves the formula for the area of a rectangle, and the formula for the volume of a cylinder involves the formula for the area of a circle.

PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photograph. Ask students to suggest possible connections between the photo and the subject of this lesson, the volume of prisms and cylinders. Then preview the Lesson Performance Task.

right cylinder right prism

right prism

© Houghton Mifflin Harcourt Publishing Company

Essential Question: How do the formulas for the volume of a prism and a cylinder relate to area formulas that you already know?

A

B

axis

right cylinder

On a sheet of paper draw a quadrilateral shape. Make sure the sides aren’t parallel. Assume the figure has an area of B square units.

area is B square units

Use it as the base for a prism. Take a block of Styrofoam and cut to the shape of the base. Assume the prism has a height of 1 unit. height is 1 unit

How would changing the area of the base change the volume of the prism?

An increase in the area of the base would increase the volume. A decrease in the area of the base would decrease the volume. Module 21

Lesson 1

1121

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made throu

be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction

Date Class Name

e of 21.1 Volum

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Prisms an

Cylinders

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HARDCOVER PAGES 11211132

Resource Locker

Quest Essential

COMMON CORE

ing a

Develop

Explore

IN2_MNLESE389847_U9M21L1.indd 1121

number of figure is the the 1 cubic unit dimensional in the interior of Volume = ned e of a threea volume units contai The volum each with ing cubic up of 8 cubes, 8 cubic centimeters. nonoverlapp prism is made has a volume of las. figure. This so it centimeter, develop volume formu la of 1 cubic to this idea volume formu ar to it You can use develop a e how to perpendicul are explor that y you’ll er. er has bases In this activit a right cylind A right cylind prism and ndicular for a right center axis. are perpe edges that gles. has lateral all rectan A right prismwith faces that are to the bases,

Watch for the hardcover student edition page numbers for this lesson.

axis axis er

er prism cylind right right

right cylind

right prism

© Houghto

n Mifflin

Harcour t

Publishin

y g Compan

l shape. Make a quadrilatera figure has an e the of paper draw el. Assum On a sheet aren’t parall sure the sides units. square area of B a block of a prism. Take Assume the base for of the base. Use it as the shape and cut to Styrofoam of 1 unit. e the has a height the prism the base chang the area of changing How would prism? d the of e volum base woul area of the of ase in the in the area An incre ase e. A decre the volum e. increase the volum ase would decre the base 1121 Module 21

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47_U9M2

ESE3898

IN2_MNL

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Lesson 21.1

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area is B square

units

height is

1 unit

Lesson 1

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8:00 AM

4/19/14 7:58 AM

If the base has an area of B square units, how many cubic units does the prism contain?

B cubic units

EXPLORE

Now use the base to build a prism with a height of h units.

Developing a Basic Volume Formula INTEGRATE TECHNOLOGY Students have the option of doing the Explore activity either in the book or online.

height is h units

QUESTIONING STRATEGIES How much greater is the volume of this prism compared to the one with a height of 1?

How are the units used to measure volume related to the units used to measure length? The units used to measure volume are cubes of the units used to measure length.

The volume of the prism is B • h, which is h times the volume of the smaller prism. Reflect

1.

2.

Suppose the base of the prism was a rectangle of sides l and w. Write a formula for the volume of the prism using l, w, and h. V = lwh

How can you estimate the volume of a cylinder whose radius and height are both 1 cm? πr 2h = π(1) 2 · 1 ≈ 3.14 cm 3

A cylinder has a circular base. Use the results of the Explore to write a formula for the volume of a cylinder. Explain what you did. V = πr 2h; I multiplied the area of a circle, πr 2, by the height of the cylinder.

Explain 1

© Houghton Mifflin Harcourt Publishing Company

The general formula for the volume of a prism is V = B ∙ h. With certain prisms the volume formula can include the formula for the area of the base.

Volume of a Prism The formula for the volume of a right rectangular prism with length ℓ, width w, and height h is V = ℓwh.

The formula for the volume of a cube with edge length s is V = s 3.

h

B

INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Begin by briefly reviewing the definitions of

Finding the Volume of a Prism

B

h W

h

l

S

S

lS WS

prism and cylinder. Be sure that students recognize the similarities in these three-dimensional figures and that they can identify the bases of a given prism or cylinder.

EXPLAIN 1

S S

Finding the Volume of a Prism Module 21

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Lesson 1

AVOID COMMON ERRORS

PROFESSIONAL DEVELOPMENT IN2_MNLESE389847_U9M21L1.indd 1122

Learning Progressions

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Previously, students saw that the formula for the area of a rectangle is the starting point for developing the area formulas for other polygons. In much the same way, the formula for the volume of a rectangular prism (V = Bh) is the starting point for developing the volume formulas for other three-dimensional figures. Another important idea in developing volume formulas is Cavalieri’s Principle, which states that if two threedimensional figures have the same height and the same cross-sectional area at every level, then they have the same volume. As students progress through more advanced courses, such as calculus, they will apply Cavalieri’s Principle to more complex solid figures.

Students may have difficulty finding the volume of a prism if they need to use conversion factors. Review how to use unit analysis and conversion factors to find volume.

Volume of Prisms and Cylinders 1122

Example 1

QUESTIONING STRATEGIES

How would you find the volume of the 3 contents of a right prism that is __ 4 full? Multiply the volume found by using the 3. volume formula by __ 4

Use volume formulas to solve real world problems.

A shark and ray tank at the aquarium has the dimensions shown. Estimate the volume of water in gallons. Use the conversion 1 gallon = 0.134 ft 3. Step 1 Find the volume of the aquarium in cubic feet. V = ℓwh = (120)(60)(8) = 57,600 ft 3

120 ft

8 ft

1 gallon Step 2 Use the conversion factor _3 to estimate 0.134 ft the volume of the aquarium gallons.

60 ft

1 gallon 1 gallon _ 57,600 ft 3 ∙ _3 ≈ 429,851 gallons =1 0.134 ft 0.134 ft 3 1 gallon Step 3 Use the conversion factor _ to estimate the weight of the water. 8.33 pounds 8.33 pounds 429,851 gallons ∙ __ ≈ 3,580,6549 pounds 1 gallon

8.33 pounds __ =1 1 gallon

The aquarium holds about 429,851 gallons. The water in the aquarium weighs about 3,580,659 pounds.

Chemistry Ice takes up more volume than water. This cubic container is filled to the brim with ice. Estimate the volume of water once the ice melts.

ge AB

Density of ice: 0.9167 g/cm 3 Density of water: 1 g/cm 3 Step 1 Find the volume of the cube of ice.

© Houghton Mifflin Harcourt Publishing Company

V = s 3 = 3 3 = 27 cm 3

Step 2 Convert the volume to mass using the conversion factor g 0.9167 _3 . cm g 27 cm 3 ∙ 0.9167 _3 ≈ 24.8 g cm Step 3 Use the mass of ice to find the volume of water. Use the conversion factor 24.8 g ∙

cm 1___ g 3

3 cm

cm3 1_ . g

≈ 24.8 cm 3

Reflect

3.

The general formula for the volume of a prism is V = B ∙ h. Suppose the base of a prism is a parallelogram of length l and altitude h. Use H as the variable to represent the height of the prism. Write a volume formula for this prism. V=ℓ∙h∙ H

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Small Group Activity

4/19/14 7:58 AM

Have students work in groups to make cylinders out of modeling clay. Then have them use string to slice the cylinders into eight congruent sectors and arrange the sectors to approximate a rectangular prism. Have them use this shape to explain the volume formula for a cylinder.

1123

Lesson 21.1

Your Turn

4.

Find the volume of the figure.

5.

EXPLAIN 2

Find the volume of the figure.

Finding the Volume of a Cylinder QUESTIONING STRATEGIES

72k 3 cubic units

88 cubic units

Explain 2

How is an oblique cylinder similar to a right cylinder? The bases are circles. The bases are connected by a curved lateral surface. The same volume formula works for both types of cylinders.

Each cube has a side of 2k.

Volume = 8 cubic units

Finding the Volume of a Cylinder

How is an oblique cylinder different from a right cylinder? In an oblique cylinder, the axis is not perpendicular to the base.

You can also find the volume of prisms and cylinders whose edges are not perpendicular to the base.

Oblique Prism

Oblique Cylinder

An oblique prism is a prism that has at least one non-rectangular lateral face.

An oblique cylinder is a cylinder whose axis is not perpendicular to the bases.

h

h h

h

© Houghton Mifflin Harcourt Publishing Company

Cavalieri’s Principle If two solids have the same height and the same cross-sectional area at every level, then the two solids have the same volume.

h

h

Module 21

h

h

1124

Lesson 1

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Kinesthetic Experience

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Kinesthetic learners can use a stack of pennies to understand Cavalieri’s Principle. Have students arrange the pennies to form a right cylinder and ask them to estimate the volume. Then have students push the stack to form an oblique cylinder. Students should see that the volume of the stack does not change, because the number and size of the pennies have not changed. This is supported by Cavalieri’s Principle because the cross-sectional area at each level (that is, the area of the face of a penny) is unchanged when the stack is pushed to form the oblique cylinder.

Volume of Prisms and Cylinders 1124

Example 2

AVOID COMMON ERRORS Students may make careless errors if they do not read the problem carefully and use the given information correctly. For example, if the diameter of a cylinder is given, they must divide by 2 to find the radius before using the formula. Also, the base area may be given instead of the radius.

To find the volume of an oblique cylinder or oblique prism, use Cavalieri’s Principle to find the volume of a comparable right cylinder or prism.

The height of this oblique cylinder is three times that of its radius. What is the volume of this cylinder? Round to the nearest tenth. Use Cavalieri’s Principle to find the volume of a comparable right cylinder. Represent the height of the oblique cylinder: h = 3r Use the area of the base to find r: πr 2 = 81π cm 2 Calculate the height: h = 3r = 27 cm Calculate the volume: V = Bh = (81π)27 ≈ 6870.7

B = 81π cm2

The volume is about 6870.7 cubic centimeters.

The height of this oblique square-based prism is four times that of side length of the base. What is the volume of this cylinder? Round to the nearest tenth. Calculate the height of the oblique cylinder: h = 4 s, where s is the length of the square base. Calculate the volume.

Use the area of the base to find s.

V = Bh

s 2 = 75 cm 2

= (75 cm 2)

_

s=

√ 75

cm

© Houghton Mifflin Harcourt Publishing Company

―

√75

_

)

B = 75 cm2

4√ 75 cm

= 2598.1 cm 3

Calculate the height. h = 4s = 4

(

cm

Your Turn

Find the volume. 6.

7. r = 12 in. h = 45 in.

h = (x + 2) cm 5x cm 4x cm

V = πr 2h = π(12 in.) (34 in.) = 6480 in. 3 2

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V = (4x)(5x)(x + 2) = 20x 2(x + 2)

Lesson 1

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Connect Vocabulary To help students remember the vocabulary in the lesson, including right prism, right cylinder, oblique prism, and oblique cylinder, have students make a small poster showing examples of each solid figure. Then have them use colored pencils to mark the dimensions of the base of each in one color, the heights in another color, and then list the formulas for volume in different colors. Have them label the figures with the vocabulary words, and display the posters in the classroom. Invite students to present their posters to the class.

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Lesson 21.1

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Finding the Volume of a Composite Figure

Explain 3

EXPLAIN 3

Recall that a composite figure is made up of simple shapes that combine to create a more complex shape. A composite three-dimensional figure is formed from prisms and cylinders. You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure. Example 3

Finding the Volume of a Composite Figure

5 ft

Find the volume of each composite figure.

Find the volume of the composite figure, which is an oblique cylinder on a cubic base. Round to the nearest tenth.

INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Students should be able to recognize the solid

h = 22 ft

The base area of the cylinder is B = πr 2 = π(5) = 25π ft 2. 2

The cube has side lengths equal to the diameter of the cylinder’s circular base: s = 10. The height of the cylinder is h = 22 - 10 = 12 ft.

figures that make up a composite figure. Ask them to make an organized list of all figures comprising the composite figures and then give the formulas for their volumes. Emphasize that while the volume of each cylinder or prism has the formula V = Bh, where B is the area of the base and h is the height, finding B changes from one solid figure to the next, depending on the figure. After organizing how to find the volume of the composite figure, have students substitute for the variables in each formula, find the volume of each solid, and then add the volumes to get the total.

The volume of the cube is V = s 3 = 10 3 = 1000 ft 3. The volume of the cylinder is V = Bh = (25π ft 2)(12 ft) ≈ 942.5 ft 3. The total volume of the composite figure is the sum of the individual volumes. V = 1000 ft 3 + 942.5 ft 3 = 1942.5 ft 3

This periscope is made up of two congruent cylinders and two congruent triangular prisms, each of which is a cube cut in half along one of its diagonals. The height of each cylinder is 6 times the length of the radius. Use the measurements provided to estimate the volume of this composite figure. Round to the nearest tenth. Use the area of the base to find the radius. B = πr 2

πr 2 = 36 π, so r =

6 in.

h = 6r = 6 ∙ 6 = 36 in. The faces of the triangular prism that intersect the cylinders are congruent squares. The length of the sides of each square are the same as the diameter of the circle. s=d=2∙

6 = 12 in.

The two triangular prisms form a cube. What is the volume of this cube? 3

V = s 3 = 12 =

1728

in 3

Find the volume of the two cylinders: V = 2 ∙ 36π ∙

6 =

432π

© Houghton Mifflin Harcourt Publishing Company

B = 36π in.2

Calculate the height of the cylinder:

QUESTIONING STRATEGIES How can you find the volume of a composite figure? You can divide a composite figure into component figures, use the volume formula for each component figure, then add the individual volumes.

in 3

The total volume of the composite figure is the sum of the individual volumes. V=

1728

Module 21

IN2_MNLESE389847_U9M21L1.indd 1126

in 3 +

432π

in 3 ≈ 3085.17 in 3 1126

Lesson 1

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Volume of Prisms and Cylinders 1126

Reflect

ELABORATE

8.

QUESTIONING STRATEGIES How do you find the volume of a prism or cylinder? Multiply the area of the base of the figure by the height.

A pipe consists of two concentric cylinders, with the inner cylinder hollowed out. Describe how you could calculate the volume of the solid pipe. Write a formula for the volume. Find the volume of the large cylinder and subtract the volume of

r1

r2

the smaller cylinder. If r 1 is the radius of the larger cylinder and r 2

h

is the radius of the smaller cylinder and h is the common height, this is the volume of the solid pipe. V = πr 1 2h - πr 2 2h = πh(r 1 2 - r 2 2)

SUMMARIZE THE LESSON

\ Your Turn

How is the formula for the volume of a prism similar to the formula for the volume of a cylinder? How are the formulas different? Both of the formulas may be written as V = Bh, where B is the area of the base and h is the height, but in the formula for the volume of a cylinder, B represents a circular area, while in the formula for the volume of a prism, B represents the area of a polygon.

9.

This robotic arm is made up of two cylinders with equal volume and two triangular prism hands. The volume of each hand is __12 r × __13 r × 2r, where r is the radius of the cylinder’s base. What fraction of the total volume do the robotic hands take up? 2 r3 V Arms = _ 3 2 r 3 = 2r 2 ⋅ πh + _ 1r 2 r 3 = 2πr 2h + _ V Total = 2(πr 2 ⋅ h) + _ 3 3 3 2 _r 3 V Arms 3 r r _ =_ =_ = __ V Total 3πh + r 1r 1r 2r 2 πh + _ 3 πh + _ 3 3

(

(

)

(

)

)

Elaborate

© Houghton Mifflin Harcourt Publishing Company

10. If an oblique cylinder and a right cylinder have the same height but not the same volume, what can you conclude about the cylinders? They have different radii. 11. A right square prism and a right cylinder have the same height and volume. What can you conclude about the radius of the cylinder and side lengths of the square base?

Let V 1 be the volume of the prism. Let V 2 be the volume of the cylinder. V1 = V2 s ∙ h = πr 2 ∙ h 2

s = πr 2 2

s=

―∙r

√π

12. Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder? Sample answer: I can multiply the expression for the area of a circle by the height of the

cylinder to get an expression for the volume of the cylinder.

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Evaluate: Homework and Practice 1.

EVALUATE • Online Homework • Hints and Help • Extra Practice

The volume of prisms and cylinders can be represented with Bh, where B represents the area of the base. Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula.

( )

A. V = (πr 2)h

B. V = _12 bh h

C. V = ℓwh

ASSIGNMENT GUIDE rectangular prism

triangular prism

cylinder

rectangular prism

cylinder

triangular prism

C. V = ℓwh

A. V = (πr ) ∙ h

1 B. V = _ bh ∙ h 2

( )

2

Find the volume of each prism or cylinder. Round to the nearest hundredth. 2.

3. 5.6 mm 15 yd

3.5 mm

8.4 mm

V = (8.4 mm)(3.5 mm)(5.6 mm)

9 yd

(

12 yd

)

1 ∙ 9 yd ∙ 15 yd 12 yd V= _ 2

= 164.64 mm 3

4.

(

)

54 m 2. The area of the hexagonal base is ______ tan 30° Its height is 8 m.

V = Bh =

54 m (_____ tan 30° )

2

5.

∙8m

(

)

125 The area of the pentagonal base is _____ m 2. tan 36° Its height is 15 m.

V = Bh =

≈ 748.25 m 3

125 m ∙ 15 m (_ tan 36° ) 2

≈ 2580.72 m 3

6.

7. 6 cm

9 cm

Exercise 1

Example 1 Finding the Volume of a Prism

Exercises 2–6

Example 2 Finding the Volume of a Cylinder

Exercises 7–8

Example 2 Finding the Volume of a Composite Figure

Exercises 9–12

12 ft

= 216 cm 3

V = π ∙ (6 ft) ∙ 10 ft 2

Module 21

Exercise

Explore Developing a Basic Volume Formula

approach to finding the volume of a prism. Have students imagine an arrangement of 3 rows of 4 unit cubes each. The cubes form a 3 × 4 × 1 rectangular prism with a volume of 12 cubic units. To find the volume of a 3 × 4 × 2 prism, imagine another layer of cubes stacked on top of the first one.

10 ft

4 cm

V = 9 cm ∙ 4 cm ∙ 6 cm

IN2_MNLESE389847_U9M21L1.indd 1128

© Houghton Mifflin Harcourt Publishing Company

= 810 yd 3

Practice

INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Some students may benefit from a hands-on

= (67.5 yd )12 yd 2

Concepts and Skills

1128

Depth of Knowledge (D.O.K.)

≈ 1130.97 ft 3

COMMON CORE

Lesson 1

Mathematical Practices

1–12

1 Recall of Information

MP.5 Using Tools

13–15

2 Skills/Concepts

MP.4 Modeling

16

2 Skills/Concepts

MP.5 Using Tools

17–18

2 Skills/Concepts

MP.1 Problem Solving

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Volume of Prisms and Cylinders 1128

8.

AVOID COMMON ERRORS When finding the volume of composite figures, some students may think they need to subtract the areas of the common surfaces that are shared by the touching cubes, as they would if they were finding the surface area of a composite figure. Using concrete models, show them that the volume of a composite figure made up of stacked blocks is the sum of the volumes of the individual blocks.

Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown. What is the volume of the base? Round to the nearest thundredth. (Hint: Use the right triangle in the cylinder to find its height.) a 2 + 142 = 172 a + 196 = 289

―

To remember how the volume of a prism is connected to the volume of familiar figures like cubes and boxes, have students make a graphic organizer listing the volume formulas they know and adding the volume formulas from this module. Sample:

4 ft

Cylinder

V = πr 2h

V = Bh

V = lwh

10.

Prism

Cylinder

V = π ∙ (4 ft) ∙ 4 ft

5 in.

≈ 201.06 ft

Large cylinder

V = π ∙ (10 in) ∙ 15 in

V = 12 ft ∙ 6 ft ∙ 14 ft

2

≈ 1008 ft

3

2

≈ 4712.39 in

3

201.06 ft 3 + 1008 ft 3 = 1209.06 ft 3

V ≈ 1209.1 ft

V = Bh 11.

3

4 cm

12. The two figures on each end combine to form a right cylinder.

4 cm

6 cm 8 cm

6 cm

4 ft 4 ft

V = π ∙ (2 ft) ∙ 4 ft 2

≈ 50.27 ft

3

3

Prism

V = (12 ft) ∙ (4 ft) ∙ (4 ft)

3

= 192 ft 3

50.27 ft 3 + 192 ft 3 = 242.27 ft 3

64 cm 3 + 216 cm 3 + 512 cm 3 = 792 cm 3 Module 21

2 ft

One whole cylinder

3

= 512 cm 3

Exercise

≈ 1178.10 in 3

12 ft

= 64 cm 3

IN2_MNLESE389847_U9M21L1.indd 1129

2

2 ft

8 cm 8 cm

V = (6 m )

V = π ∙ (5 in) ∙ 15 in

V = 4712.39 in 3 - 1178.10 in 3 = 3534.29 in 3

6 cm

V = (4 cm)

Small cylinder

3

4 cm

V = (8 cm)

Lesson 21.1

10 in.

15 in.

= 216 cm 3

1129

3

12 ft

© Houghton Mifflin Harcourt Publishing Company

Box (prism)

V = Bh

1 liter ≈ 1.48 liters (_ 1000 cm )

14 ft

Volume

V = s3

≈ 1484.52 cm

1484.52 cm 3 ∙

The volume is 1.5 liters.

3

6 ft 4 ft

Cube

―

2 = π ∙ (7 cm) ∙ √93 cm

Find the volume of each composite figure. Round to the nearest tenth.

CONNECT VOCABULARY

Volume

14 cm

= π r 2h

a = √93 cm 3

9.

Shape

V = Bh

2

a2 = 93

17 cm

Lesson 1

1129

Depth of Knowledge (D.O.K.)

COMMON CORE

Mathematical Practices

19

3 Strategic Thinking

MP.1 Problem Solving

20

3 Strategic Thinking

MP.4 Modeling

21

3 Strategic Thinking

MP.2 Reasoning

4/19/14 7:58 AM

13. Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle. If he wants to fill it to a depth of 4 in., how many cubic yards of dirt does he need? Round to the nearest cubic yard. If dirt costs $25 per yd 3, how much will the project cost? _ 1 yd 1 yd 1 yd 4 ft = 48 ft 3; 48 ft 3 ∙ V = Bh = ℓwh = 9 ft ∙ 16 ft ∙ ∙ ∙ = 1.7 ≈ 2 yd 3; 12 3 ft 3 ft 3 ft $25 2 yd 3 ∙ = $50; 2 yd 3; $50 yd 3

(_ ) (_ ) (_ )

(_ )

_

INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 When working with the volume of a cylinder

a. Find the height h of the container to the nearest tenth.

formulas, remind students that if they use 3.14 instead of the π key on a calculator, they are automatically giving an approximation to the volume of a cylinder. Have small groups of students discuss when using the π key on a calculator is appropriate and when it is not necessary.

1 in.

14. Persevere in Problem Solving A cylindrical juice container with a 3 in. diameter has a hole for a straw that is 1 in. from the side. Up to 5 in. of a straw can be inserted.

5 in.

―

h

a2 + b 2 = c 2; h2 + 2 2 = 5 2; h 2 + 4 = 25; h 2 = 21; h = √21 ; h ≈ 4.6 in.

b. Find the volume of the container to the nearest tenth.

3 in.

V = Bh = πr 2h = π ∙ (1.5 in) ∙ 4.6 in. ≈ 32.5 in 3 2

How many ounces of juice does the container hold? (Hint: 1 in 3 ≈ 0.55 oz)

c.

32.5 in 3 ∙

0.55 oz _ ≈ 17.9 oz

15. Abigail has a cylindrical candle mold with the dimensions shown. If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm, about how many candles can she make after melting the block of wax? Round to the nearest tenth.

Wax

Mold

V = Bh

V = Bh

= ℓwh

= 15 cm ∙ 12 cm ∙ 18 cm = 3240 cm

INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 If the corresponding linear dimensions of

1 in 3

3

3240 cm ÷ 218 cm ≈ 14.9 3

3.4 cm

two similar solid figures have a scale factor of k, then their volumes are in the ratio of 1:k 3. For example, if the dimensions of a cube with an edge length of 4 cm are doubled, then the volumes are in the ratio 1:2 3, or 1:8. This means, the volume goes from 4 3 = 64 cm 3 to 8 3 = 512 cm 3.

6.0 cm

= πr 2h

= π ∙ 3.4 2 ∙ 6 ≈ 218 cm 3

3

x+1

16. Algebra Find the volume of the three-dimensional figure in terms of x.

V = Bh

= πr 2h

x

= π ∙ (x + 1) ∙ x 2

= π ∙ (x + 2x + 1) ∙ x 2

= π ∙ (x 3 + 2x 2 + x) = πx 3 + 2πx 2 + πx

© Houghton Mifflin Harcourt Publishing Company

Because 0.9 of a candle would not make an entire candle, the answer rounds down to 14 candles.

V = πx 3 + 2πx 2 + πx

Module 21

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Lesson 1

4/19/14 7:58 AM

Volume of Prisms and Cylinders 1130

17. One cup is equal to 14.4375 in 3. If a 1-cup measuring cylinder has a radius of 2 in., what is its height? If the radius is 1.5 in., what is its height? Round to the nearest tenth. 2 in. radius 1.5 in. radius

PEERTOPEER DISCUSSION Have students work in pairs. Each student should make up four application problems, two involving the volumes of prisms and two involving the volumes of cylinders. Then have them exchange problems with their partners, and solve the problems.

V = Bh

V = Bh

V = πr 2h

V = π r 2h

14.4375 in 3 = π(1.5 in.) h

14.4375 in = π2 h 3

2

2

14.4375 in __ =h

14.4375 in __ =h

1.1489 in. ≈ h

2.0425 in. ≈ h

3

3

π(2 in.)

π(1.5 in.)

2

JOURNAL Have students describe how to use the general volume formula V = Bh to write a formula for the volume of a rectangular prism with base length ℓ and width w, and for a cylinder with radius r.

18.

2

h ≈ 1.2 in. h ≈ 2.0 in. Make a Prediction A cake is a cylinder with a diameter of 10 in. and a height of 3 in. For a party, a coin has been mixed into the batter and baked inside the cake. The person who gets the piece with the coin wins a prize. a. Find the volume of the cake. Round to the nearest tenth.

V = π(5 in.) (3 in.) ≈ 235.6 in. 3 2

b. Keka gets a piece of cake that is a right rectangular prism with a 3 in. by 1 in. base. What is the probability that the coin is in her piece? Round to the nearest hundredth. 9 in 3 V = (3 in.)(1 in.)(3 in.) = 9 in. 3; Probability = ≈ 0.04 235.6 in 3

_

H.O.T. Focus on Higher Order Thinking

19. Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views?

Top prism: V = 10 cm ∙ 6 cm ∙ 4 cm = 240 cm

3

© Houghton Mifflin Harcourt Publishing Company

4 cm 10 cm 10 cm Side

Top

V = 840 cm 3

20. Draw Conclusions You can use displacement to find the volume of an irregular object, such as a stone. Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in. A stone is placed in the tank so that it is completely covered, causing the water level to rise by 2 in. Find the volume of the stone.

Before the stone: V = 2 ft ∙ 1 ft ∙

8 4 _ ft = _ ft , or V = 24 in. ∙ 12 in. ∙ 8 in. = 2304 in 12

_ _ _ _

3

3

3

_

10 5 After the stone: V = 2 ft ∙ 1 ft ∙ ft = ft 3, or V = 24 in. ∙ 12 in. ∙ 10 in. = 2880 in 3 12 3 5 4 1 Volume of the stone = - = ft 3, or 2880 - 2304 = 576 in 3 3 3 3

Module 21

IN2_MNLESE389847_U9M21L1.indd 1131

Lesson 21.1

4 cm 10 cm Front

Bottom prism: V = 10 ∙ 10 ∙ 6 = 600 cm 3

1131

10 cm

1131

Lesson 1

4/19/14 7:58 AM

21. Analyze Relationships One juice container is a rectangular prism with a height of 9 in. and a 3 in. by 3 in. square base. Another juice container is a cylinder with a radius of 1.75 in. and a height of 9 in. Describe the relationship between the two containers.

Prism

INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Discuss with the class different methods of

Cylinder V = Bh

V = Bh

= ℓwh

= π r 2h

= 81 in 3

≈ 86.6 in 3

= 3 in. ∙ 3 in. ∙ 9 in.

calculating the volume of paper towels in the Lesson Performance Task. The solution accompanying the task is to subtract the volume of a 1-inch-radius cylinder from the volume of a 3-inch-radius cylinder:

= π ∙ (1.75 in.) ∙ 9 in. 2

The cylinder’s volume is greater than the rectangular prism’s volume by 5.6 in 3.

π(3 2)(11) - π(1 2)(11) = 3.14(9)(11) - 3.14(1)(11)

Lesson Performance Task A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches.

2 in.

2 in.

Ask students to suggest a shortcut for evaluating the expression on the right, one that reduces the number of products that must be found. The key is to factor the expression:

2 in.

1. Find the volume of the paper on the roll. Explain your method. 2. Each sheet of paper on the roll measures 11 inches by 1 11 inches by __ 32 inches. Find the volume of one sheet. Explain how you found the volume.

3.14(9)(11) - 3.14(1)(11) = (9 - 1)(3.14)(11) = 8(3.14)(11)

11 in.

3. How many sheets of paper are on the roll? Explain.

INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Some students may be skeptical of the

1. The volume of paper on the roll equals the volume of the larger 3-in. radius cylinder minus the volume of the smaller 1-in. radius cylinder. Volume of outer roll: πr 2h = 3.14(3) 11

© Houghton Mifflin Harcourt Publishing Company

2

= 310.86 in 3

Volume of inner roll: πr 2h = 3.14(1) 11 2

= 34.54 in 3

Volume of paper: 310.86 - 34.54 = 276.32 in 3 2. Each sheet is a rectangular prism measuring 11 in. by 11 in. 1 by __ in. 32 1 V = 11 × 11 × __ 32

≈ 3.78 in 3

method used to calculate the number of paper towel sheets, because it may seem to erroneously convert a three-dimensional measurement—the volume of the paper towel roll—into a two-dimensional one—a flat sheet of 73 towel sections. Besides pointing out that the sheet of towel sections, while very thin, is in fact three-dimensional, mention that once the volume of the paper towel roll has been calculated, it can be used to find the parameters of any substance with that same volume, for example, the weight of water that would fill a container shaped like the paper towel roll.

3. 73 sheets; The number of sheets is the total volume of paper divided by the volume of each sheet. 276.32 ÷ 3.78 ≈ 73.1

Module 21

1132

Lesson 1

EXTENSION ACTIVITY IN2_MNLESE389847_U9M21L1.indd 1132

Students should work in teams of two or more. Each team should have a new roll of paper towels. Direct teams to find the volume of paper on the roll by subtracting the volume of the inside tube from the volume of the entire roll. Then, using the number of sheets in the roll, a figure that will be given on the outside of the package, they should determine the thickness of the paper. If teams work with different brands of towels, they can compare thicknesses, which affect the ability of a towel to absorb moisture.

4/19/14 7:58 AM

Scoring Rubric 2 points: The student’s answer is an accurate and complete execution of the task or tasks. 1 point: The student’s answer contains attributes of an appropriate response but is flawed. 0 points: The student’s answer contains no attributes of an appropriate response.

Volume of Prisms and Cylinders 1132

LESSON

21.2

Name

Volume of Pyramids

Class

Date

21.2 Volume of Pyramids Essential Question: How do you find the volume of a pyramid?

Common Core Math Standards The student is expected to: COMMON CORE

Resource Locker

G-GMD.A.1

Explore

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Also G-GMD.A.2, G-GMD.A.3, G-MG.A.1

Developing a Volume Formula

You can think of irregular pyramids as parts of a rectangular prism. This cube can be divided into three square pyramids.

Mathematical Practices COMMON CORE

MP.2 Reasoning

Language Objective Explain to a partner how to apply the formulas for the volume of a pyramid. The volume of a pyramid is related to the volume of a prism with the same base and height. This triangular pyramid can be thought of as part of a triangular prism.

ENGAGE

A E

Essential Question: How do you find the volume of a pyramid?

PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photograph. Ask students to explain why the long handle cannot simply be pulled directly through the rings and away from the puzzle. Then preview the Lesson Performance Task.

F

D © Houghton Mifflin Harcourt Publishing Company

Find the area of the base and multiply it by one third times the height of the pyramid.

A

B

D

B C

C

To find the volume of the first pyramid, A-BCD, first let_ the area _of the base of △BCD _be B, and let the height of the pyramid, AD, be h. The edges EB and FC are congruent to AD and _ parallel to AD. The bases of the prism, △EFA and △BCD, are congruent.

A

What is the volume of the triangular prism in terms of B and h?

Bh; the volume of a prism is the product of the base and the height.

Module 21

be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction

Lesson 2

1133

gh “File info”

made throu

Date Class Name

e of 21.2 Volum

Pyramids

id? e of a pyram the volum area of a circle, do you find ion: How circumference G-MG.A.1 as for the G-GMD.A.3, for the formul G-GMD.A.2, al argument cone. Also Give an inform pyramid, and G-GMD.A.1 a cylinder, Formula volume of of a circle, a Volume can be eloping This cube Dev prism. gular Explore of a rectan ids as parts lar pyram irregu of ids. pyram You can think three square divided into

Resource Locker

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IN2_MNLESE389847_U9M21L2.indd 1133

base with the same e of a prism to the volum of a triangular prism. id is related as part e of a pyram can be thought of The volum id lar pyram This triangu A E

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A

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B

be B, C of △BCD _ and of the base the area _ ent to AD , first let_ are congru id, A-BCD EB and FC the first pyram be h. The edges congruent. volume of id, AD, △BCD, are To find the the pyram EFA and height of prism, △ and let the _ bases of the and h? AD. The terms of B parallel to t. prism in

What is the

© Houghto

n Mifflin

Watch for the hardcover student edition page numbers for this lesson.

.

F D

HARDCOVER PAGES 11331144

ular the heigh the triang base and volume of ct of the is the produ e of a prism

Bh; the volum

Lesson 2 1133 Module 21

1L2.indd

47_U9M2

ESE3898

IN2_MNL

1133

Lesson 21.2

1133

4/19/14

8:16 AM

4/19/14 8:15 AM

B

You will now compare the volume of pyramid A-BCD and the volume of the triangular prism. _ • Draw EC. This is the diagonal of a rectangle, so △ EBC ≅ △ CFE . • Explain why pyramids A-EBC and A-CFE have the same volume. A

EXPLORE Developing a Volume Formula

A

E

INTEGRATE TECHNOLOGY

E F

F

D

B

Students have the option of doing the Explore activity either in the book or online.

D

B C

QUESTIONING STRATEGIES

C

What information do you need to find the volume of a pyramid? You need the base area of the pyramid and the height. The volume is __13 the base area times the height.

• Explain why pyramids C-EFA and A-BCD have the same volume. A

A

E

E F

F

D

B

How does the shape of the base of the pyramid affect the volume? The shape of the base affects how the area of the base is calculated, which in turn affects the volume.

D

B C

C

The bases are congruent and the heights are equal, so by the preceding postulate, the

The bases are congruent and the heights are equal, so by the preceding postulate, the volumes are equal.

C

You have now shown that the three pyramids that form the triangular prism all have the same volume. Compare the volume of pyramid A-BCD and the volume of the triangular prism.

The volume of pyramid A-BCD of one-third the volume of the triangular prism.

D

INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Begin by brievfly reviewing the definition of

© Houghton Mifflin Harcourt Publishing Company

volumes are equal.

pyramid and the associated vocabulary (lateral face, vertex, base). You may also want to review how to sketch a pyramid. Suggest that students start by drawing a polygonal base and plotting a point for the vertex. Then, students can draw segments from the vertex to each vertex of the polygonal base. Remind students to use dashed lines for edges that are hidden when the pyramid is viewed from the front.

Write the volume of pyramid A-BCD in terms of B and h. 1 The volume of pyramid A-BCD is Bh. 3

_

Module 21

1134

Lesson 2

PROFESSIONAL DEVELOPMENT IN2_MNLESE389847_U9M21L2.indd 1134

Math Background

4/19/14 8:15 AM

In this lesson, students develop and use a formula for the volume of a pyramid. They find that the volume of a pyramid is related to the volume of a prism by a factor of one third. That is, the volume of a pyramid is equal to one-third the area of the associated prism, or one-third the area of the base of the pyramid times its height. Cavalieri’s Principle also applies to pyramids, although the application is less straightforward than it is with prisms and cylinders, because the crosssectional area of a pyramid is different at different points along its height.

Volume of Pyramids 1134

Reflect

EXPLAIN 1

Explain how you know that the three pyramids that form that triangular prism all have the same volume. Pyramids A-EBC and A-CFE have the same volume and pyramids C-EFA and

1.

Finding the Volume of a Pyramid

A-BCD have the same volume. But A-CFE and C-EFA are two names for the same pyramid, so by the Transitive Property of Equality, the pyramids all have the same volume.

QUESTIONING STRATEGIES What must you know about a line before you can use it as a height in the formula for the volume of a pyramid? The line must be perpendicular to the base and contain the vertex.

Explain 1

Finding the Volume of a Pyramid

Since the volume of any triangular or rectangular pyramid can be shown to compose one third the volume of a rectangular prism, the formula for the volume of a triangular prism can be generalized.

Volume of a Pyramid The volume V of a pyramid with base area B and height h is given by V= __13 Bh. Example 1

Solve a volume problem.

Ashton built a model square-pyramid with the dimensions shown. What is the volume of the pyramid? The pyramid is composed of wooden blocks that are in the shape of cubes. A block has the dimensions 4 cm by 4 by 4 cm. How many wooden blocks did Ashton use to build the pyramid? 16 cm

© Houghton Mifflin Harcourt Publishing Company

24 cm 24 cm

• Find the volume of the pyramid. The area of the base B is the area of the square with sides of length 24 cm. So, B = 576 cm 2. 1 Bh = _ 1 · 576 · 16. The volume V of the pyramid is _ 3 3 So V = 9,216 cm 3.

• Find the volume of an average block. The volume of a cube is given by the formula V = s 3. So the volume W of a wooden block is 64 cm 3.

• Find the approximate number of stone blocks in the pyramid, divide V by W. So the number of blocks that Ashton used is 144.

Module 21

1135

Lesson 2

COLLABORATIVE LEARNING IN2_MNLESE389847_U9M21L2.indd 1135

Small Group Activity Have students make nets for a square-based pyramid and a square-based prism that has the same height as the pyramid. Then, have students cut out, fold, and tape the nets to form the three-dimensional figures. Students can model the volume of the pyramid by filling it with uncooked rice or sand. Ask students to pour the rice from the pyramid into the prism as many times as necessary to see how the volumes of the figures are related. Students will discover that it takes three batches of rice from the pyramid to fill the prism. That is, the volume of the pyramid is one-third the volume of the associated prism.

1135

Lesson 21.2

4/19/14 8:15 AM

B

The Great Pyramid in Giza, Egypt, is approximately a square pyramid with the dimensions shown. The pyramid is composed of stone blocks that are rectangular prisms. An average block has dimensions 1.3 m by 1.3 m 146 m by 0.7 m. Approximately how many stone blocks were used to build the 230 m pyramid? Round to the 230 m nearest hundred thousand.

AVOID COMMON ERRORS When calculating volumes of pyramids, some students may forget to multiply by __13 (or divide by 3). Remind students that pyramids need the factor of __13 , but prisms and cylinders do not.

• Find the volume of the pyramid. 2 The area of the base B is the area of the square with sides of length 230 m. So, B = 52,900 m .

1 Bh = _ 1‧ The volume V of the pyramid is _ 3 3 m3 So V = V .

52, 900

146

‧

.

• Find the volume of an average block. The volume of a rectangular prism is given by the formula average block is

1.183 m

3

V = lwh

. So the volume W of an

.

• Find the approximate number of stone blocks in the pyramid, divide by

W

V

. So the approximate number of blocks is 2,200,000 .

Reflect

What aspects of the model in Part B may lead to inaccuracies in your estimate? Possible answer: The given dimensions of the pyramid and the blocks are approximations,

and the blocks do not form a true pyramid, and consist of layers or “steps”. 3.

Suppose you are told that the average height of a stone block 0.69 m rather than 0.7 m. Would the increase or decrease your estimate of the total number of blocks in the pyramid? Explain. Increase; this change would decrease W, so V would increase. Since the size of each stone

__ W

was slightly smaller, more would be needed to make a pyramid the same size. Your Turn

4.

A piece of pure silver in the shape of a rectangular pyramid with the dimensions shown has a mass of 19.7 grams. What is the density of silver? Round to the nearest tenth. mass Hint: density = _____ . volume

(

)

• Find the area of the base: B = (2.5)(1.5) = 3.75 cm 1 (3.75)( 1.5 ) = 1.875 • Find the volume of the pyramid: V = _ 3 19.7 • Find the density: D = _ = 10.5 1.875 Module 21

2

1136

1.5 cm

2.5 cm

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Mark Goddard/iStockPhoto.com

2.

1.5 cm

Lesson 2

DIFFERENTIATE INSTRUCTION IN2_MNLESE389847_U9M21L2.indd 1136

Modeling

4/19/14 8:15 AM

To help students remember how to apply the formula for the volume of a pyramid, have students make a small poster showing examples of several types of regular pyramids, including ones with square, pentagonal, and hexagonal bases. Then have them use colored pencils to write the area of each base in one color, the heights in another color, and then list the formulas for the volumes in different colors. Have them label each figure with the formula and volume, and display the posters in the classroom. Invite students to share their posters with other students.

Volume of Pyramids 1136

Explain 2

EXPLAIN 2

You can add or subtract to find the volume of composite figures. Example 2

Finding the Volume of a Composite Figure

15 ft 12 ft

25 ft

• Find the volume of the prism. V = lwh = (25)(12)( 15 ) = 4500 ft 3

figures that make up a composite figure. Ask them to make an organized list of all figures comprising the composite figures and then give the formulas for their volumes. Emphasize that the composite figure may include cylinders or prisms along with pyramids. After organizing how to find the volume of the composite figure, have students substitute for the variables in each formula, find the volume of each solid, then add or subtract the volumes to get the total.

• Find the volume of pyramid. Area of base: B = (25)( 12 ) = 300 ft 2 1 (300)(15) = 1500 ft 3 Volume of pyramid: V = _ 3 • Subtract the volume of the pyramid from volume of the prism to find the volume of the composite figure. 4500 - 1500 = 3000 So the volume of the composite figure is 3000 ft 3.

15 cm 12 cm

30 cm

© Houghton Mifflin Harcourt Publishing Company

How do you know whether to add or subtract the volumes of a composite figure? If the diagram indicates that one figure is cut out of or removed from another, subtract its volume from the volume of the larger figure. It the diagram indicates that two figures are connected to one another, add their volumes.

Find the volume of the composite figure formed by a pyramid removed from a prism. Round to the nearest tenth.

INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Students should be able to recognize the solid

QUESTIONING STRATEGIES

Finding the Volume of a Composite Figure

• Find the volume of the prism. V = lwh = (30)( 12 ) ( 15 ) = ( 5400 ) cm 3 • Find the volume of the pyramid. 2 ( )( ) Area of base: B = 30 12 = 360 cm 1 ( 360 )( 15 ) = ( 1800 ) cm 3 Volume of pyramid: V = _ 3 • Subtract volume of pyramid from volume of prism to find volume of composite figure. 5400 - 1800 = 3600 So the volume of the composite figure is 3600 cm 3.

Module 21

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Lesson 2

LANGUAGE SUPPORT IN2_MNLESE389847_U9M21L2.indd 1137

4/19/14 8:15 AM

Connect Vocabulary To help them remember how the volume of a prism is connected to the volume of a pyramid, have students make a graphic organizer listing the volume formulas they know and adding the volume formulas from this module. Sample:

Volumes of 3-Dimensional Figures

1137

Lesson 21.2

Formula

V = Bh

V = __13 Bh

Figure

prism

pyramid

Your Turn

ELABORATE

Find the volume of the composite figure. Round to the nearest tenth. 5.

The composite figure is formed from two pyramids. The base of each pyramid is a square with a side length of 6 inches and each pyramid has a height of 8 inches.

Area of base of each pyramid: B = (6)(6) = 36 in 1 (36)(8) = 96 in 2 Volume of one pyramid: V = _ 3 Volume of figure: 96 + 96 = 192 in 3 6.

QUESTIONING STRATEGIES How is the volume of a pyramid related to the volume of a prism with the same base and height? The volume of the pyramid is one-third the volume of the prism.

2

The composite figure is formed by a rectangular prism with two square pyramids on top of it. Volume of rectangular prism: V 1 = (2)(10)(5) = 100 ft 3

Base of each pyramid: (5)(5) = 25 ft 2 1 Volume of each pyramid: V 2 = (25)(3) = 25 ft 2 3 Volume of entire figure: V 1 + 2V 2 = (100) + 2(25) = 150 ft 2

_

3 ft

2 ft 10 ft

SUMMARIZE THE LESSON

5 ft

How do you find the volume of a pyramid? The volume of the pyramid is one-third the area of the base of the pyramid times the height of the pyramid.

Elaborate 7.

Explain how the volume of a pyramid is related to the volume of a prism with the same base and height. 1 the volume Three pyramids with equal volumes combine to form a prism. So a pyramid is _ 3

of the prism.

8.

If the original volume is Bh, and its length and width are doubled, then the area of the base 1 is (2ℓ)(2w), or 4ℓw, which makes the volume V = 4 × __ Bh. 3

9.

Essential Question Check-In How do you calculate the volume of a pyramid? 1 The volume, V, of a pyramid with base area B and height h is given by V = __ Bh. 3

Module 21

IN2_MNLESE389847_U9M21L2.indd 1138

1138

© Houghton Mifflin Harcourt Publishing Company

If the length and width of a rectangular pyramid are doubled and the height stays the same, how does the volume of the pyramid change? Explain. The volume of the pyramid will be 4 times as great as the volume of the original pyramid.

Lesson 2

4/19/14 8:15 AM

Volume of Pyramids 1138

Evaluate: Homework and Practice

EVALUATE 1.

Pyramid

_1 Bh 3 1 = _ℓwh

V=

Prism V = Bh

= ℓwh 3 1 The volume of the square pyramid is __ the volume of the square prism. 3

ASSIGNMENT GUIDE Concepts and Skills

Practice

Explore Developing a Volume Formula

Exercises 1–3

Example 1 Finding the Volume of a Pyramid

Exercises 4–7

Example 2 Finding the Volume of a Composite Figure

Exercises 8–9

2.

_

_1 Bh 3 _ _1 h 3 _ _h

_ _ _

© Houghton Mifflin Harcourt Publishing Company

INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 When working with the volume of a pyramid formulas, remind students that pyramids have triangles for lateral sides. Each of these triangles has its own height, but these are not the height of the pyramid. (Instead, they are used to find the surface area of the pyramid.) The height of the pyramid is the perpendicular distance from the vertex to the base.

Which of the following equations could describe a square pyramid? Select all that apply. 1 A. 3Vh = B D. V = C. V = Bh 3 3V = ℓwh 1 V _ B. V = ℓwB = 3 3V B =w 3V ℓh V C. w = _ = ℓh B 1 h V Bh E. V = _ _ D. = 3 3 B 2 1 2 h w _ = wh E. V = 3 3 w 2h 1 = VBh = F. _ 3 3

_

3.

_ _ _ _

__

Find the volume of the pyramid. Round your answer to the nearest tenth. 4.

5.

8.1 mm 15.2 mm 12.5 mm

_1 Bh 3 1 _ = (6 ‧ 4 ) (7 )

17 in.

_1 Bh 3 1 1 _ = ‧ (_ ‧ 12.5 ‧ 15.2)(8.1)

V=

3

3

= 136 in 3

2

6 in.

= 256.5 mm 3

Module 21

Exercise

IN2_MNLESE389847_U9M21L2.indd 1139

4 in. Lesson 2

1139

Depth of Knowledge (D.O.K.)

COMMON CORE

Mathematical Practices

1

2 Skills/Concepts

MP.2 Reasoning

2

2 Skills/Concepts

MP.5 Using Tools

3

2 Skills/Concepts

MP.3 Logic

4–13

1 Recall of Information

MP.5 Using Tools

14–19

2 Skills/Concepts

MP.4 Modeling

3 Strategic Thinking

MP.3 Logic

20

Lesson 21.2

3

Justify Reasoning As shown in the figure, polyhedron ABCDEFGH is a cube and P is any point on face EFGH. Compare the volume of the pyramid PABCD and the volume 1 of the cube. Demonstrate how you came to your answer. V prism = Bh V pyramid = Bh 3 = ℓwh 1 H ℓwh = G 3 =s‧s‧s P 1 = s‧s‧s E = s3 F 3 1 = s3 3 The volume of PABCD is 13 the volume C D A B of the cube.

V=

1139

• Online Homework • Hints and Help • Extra Practice

Compare the volume of a square pyramid to the volume of a square prism with the same base and height as the pyramid.

4/19/14 8:15 AM

6.

Find the volume of a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft.

7.

The area of the base of a hexagonal pyramid 24 is ______ cm 2. Find its volume.

INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 If the height of a pyramid stays the same and

tan 30°

_1 Bh 3 1 = _ ‧ 25 ‧ 9

V=

4√3 cm

_1 Bh 3 24 1 _ 3 = (_)4 √― 3 tan30° 1 _ = ‧ 288

V=

3

= 75 ft 3 4 cm

the side length of the base is doubled, then the volume of the pyramid is multiplied by the square of the dimension change, or 4. If all dimensions of the pyramid are doubled, however, then the volumes are in the ratio 1:2 3, or 1:8.

3 = 96 cm 3

Find the volume of the composite figure. Round to the nearest tenth. 8. 18 cm

V prism = Bh = 12

9.

= 1728 cm 3 1 V pyramid = Bh 3 12 cm 1 = (12 ‧ 12)(18) 3 = 864 cm 3

_ _

12 cm 12 cm

5 cm

V prism = Bh = (25 ‧ 12.5)(5) = 1562.5 cm 3 1 1 V pyramid = Bh = (12.5 ‧ 12.5)(7.5) = 390.625 cm 3 3 3

_

1728 + 864 = 2592

_1 Bh 3 1 V = _ℓwh 3 1 3969 = _ ‧ x ‧ x ‧ 21 3 1 3969 = _ ‧ x ‧ 21 3

2

_

1562.5 + 390.625 + 390.625 = 2343.75 V total = 2343.8 cm 3

11. Consider a pyramid with height 10 feet and a square base with side length of 7 feet. How does the volume of the pyramid change if the base stays the same and the height is doubled?

3969 = 7x 2

The volume

Volume of pyramid

― √―― 567 = √x

doubles.

with height doubled

Volume of pyramid

23.8 ≈ x

V=

with height 10

23.8 ft

V=

567 = x

2

2

_1 Bh 3 1 = _ℓwh 3 1 = _ ‧ 7 ‧ 7 ‧ 10

_1 Bh 3 1 = _ℓwh 3 1 = _ ‧ 7 ‧7 ‧ 20 3 _ = 326.6 in 3

© Houghton Mifflin Harcourt Publishing Company

10. Given a square pyramid with a height of 21 ft and a volume of 3969 cubic feet, find the length of one side of the square base. Round to the nearest tenth.

12.5 cm

25 cm

V total = 2592.0 cm 3

V=

7.5cm

3

3 _ = 163.3 in 3 _ _ 326.6 is twice as great as 163.3.

Module 21

Exercise

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Lesson 2

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Depth of Knowledge (D.O.K.)

COMMON CORE

Mathematical Practices

21

3 Strategic Thinking

MP.6 Precision

22

3 Strategic Thinking

MP.6 Precision

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Volume of Pyramids 1140

12. Algebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters.

AVOID COMMON ERRORS When finding the volume of composite figures, some students may think they need to always add the volumes of the figures that make up the composite figure. Caution students to study the composite figure carefully and then decide which volumes are added and which are subtracted.

13. Find the height of a rectangular pyramid with length 3 meters, width 8 meters, and volume 112 cubic meters.

_1 Bh 3 1 V = _ℓwh 3 1 112 = _ ‧ 3 ‧ 8 ‧ h V=

X

10 cm

3 112 = 8h

10 cm

_1 Bh 3 1 V = _ℓwh 3 1 200 = _ ‧ 10 ‧ 10 ‧ x 3 100 _ ‧x 200 = 3 100 _ 3 3 _ _ ‧ 200 = x‧

8h 112 _ _ =

V=

100

3

8 8 14 = h

h = 14 m

100

6=x x = 6 cm

V prism = Bh = (25 ‧ 12.5)(5) = 1562.5 cm 3 1 1 V pyramid = Bh = (12.5 ‧ 12.5)(7.5) = 390.625 cm 3 3 3

_

_

1m

14. A storage container for grain is in the shape of a square pyramid with the dimensions shown.

1562.5 + 390.625 + 390.625 = 2343.75

a. What is the volume of the container in cubic centimeters? 1 1 1 V = Bh = ℓwh = ‧ 100 ‧100 ‧150 = 500, 000 cm 3 3 3 3

_

V total = 2343.8 cm 3

_

_

1.5 m

b. Grain leaks from the container at a rate of 4 cubic centimeters per second. Assuming the container starts completely full, about how many hours does it take until the container is empty?

500, 000 cm 1 min 1 hour __ = 125, 000 sec; 125, 000 sec ‧ _ ‧ _ ≈ 34.7 hours

© Houghton Mifflin Harcourt Publishing Company

3

15. A piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of 16.76 grams. What is the density of copper? Round to the nearest hundredth. mass . Hint: density = _ volume

(

IN2_MNLESE389847_U9M21L2.indd 1141

Lesson 21.2

)

V=

Module 21

1141

60 sec

4 cm 3 per sec

60 min

1.5 cm

1.5 cm 2.5 cm

19.7 g g _1 Bh = _1 ℓwh = _1 ‧ 2.5 ‧ 1.5 ‧ 1.5 = 1.875 cm _ ≈ 10.5 _ 3

3

3

3

1.875 cm 3

1141

cm 3

Lesson 2

4/19/14 8:15 AM

16. Represent Real World Problems An art gallery is a 6 story square pyramid with base area __12 acre (1 acre = 4840 yd 2, 1 story ≈ 10 ft). Estimate the volume in cubic yards and cubic feet.

_1 Bh 3 1 1 = _(_ ‧ 4840)‧ 20

PEERTOPEER DISCUSSION Have students work in pairs. Each student makes up two application problems, one involving the volume of a pyramid and one involving the volume of a composite figure. Then students exchange problems with their partners, and solve the problems that their partners have written.

_1 Bh 3 1 1 = _(_ ‧ 4840 ‧ 3 ‧ 3)‧ 60

V=

V=

3 2 = 435,600

3 2 _ = 16,133.3

≈ 436,000 ft 3

≈ 16,100 yd 3

17. Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6? Explain how you found your answer.

4 ft

14112 _ _ = 216

Volume of original

Volume of pyramid with

pyramid

dimensions multiplied by 6

_1 Bh 3 1 _ = (7 ‧ 7)(4)

_1 Bh 3 1 _ = (42 ‧ 42)(24)

V=

V=

3 _ = 65.3 ft 3

3 = 14, 112 ft 3

18. Geology A crystal is cut into a shape formed by two square pyramids joined at the base. Each pyramid has a base edge length of 5.7 mm and a height of 3 mm. What is the volume of the crystal to the nearest cubic millimeter?

7 ft

7 ft

65.3

The volume would be 216 times larger; dividing the volume of the enlarged pyramid by the volume of the original pyramid gives 216.

19. A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards. What is the volume of the attic in cubic feet? In cubic yards? 5 yd

3 mm

_1 Bh 3 1 = _(5.7 ‧ 5.7)(3) 3 1 = _ ‧ 97.47

V=

3 = 32.49 mm 3

2 ‧ge07sec10l07003a 32.49 = 64.98

V=

_1 Bh = _1 (45 ‧ 45)(15) = 10, 125 ft

V=

_1 Bh = _1 (15 ‧ 15)(5) = 375 ft

3

3

3

3

3

3

© Houghton Mifflin Harcourt Publishing Company

45 ft 5.7 mm

AB

V ≈ 65 mm 3

ge07se_c10l07004a Module 21

IN2_MNLESE389847_U9M21L2.indd 1142

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Lesson 2

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Volume of Pyramids 1142

H.O.T. Focus on Higher Order Thinking

JOURNAL

20. Explain the Error Describe and correct the error in finding the volume of the pyramid

Have students describe how to use the general volume formula V = Bh to write a formula for the volume of a square pyramid with base length ℓ and height h.

V = 1 (49)(10) 2 = 245 ft3 10 ft

7 ft 1 1 The formula for the volume of a pyramid is V = __ Bh, not V = __ Bh. The 3 2 1 1 product of 49 and 10 should be multiplied by __ , rather than __ . 3 2

21. Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft. The volume of the pyramid is 60 ft 3. Explain how to find the length of a side of the pyramid’s base.

Let s be the length of a side of the pyramid’s base. Then the area of the 1 2( ) base is s 2, and __ s 5 = 60. Solving shows that s = 6 ft. 3 1 2( ) s 5 = 60 3 5 2 s = 60 3 3 3 5 2 ‧ s = 60 ‧ 5 3 5 s 2 = 36

_

_ _ _

_

―

―

© Houghton Mifflin Harcourt Publishing Company

√s 2 = √36 s=6

22. Critical Thinking A triangular pyramid has a length of 2, a width of x, and a height of 3x. It Its volume is 512 cm 3. What is the area of the base? 1 V = Bh 3 1 512 = (2)(x)(3x) 3 1 512 = (6x 2) 3

_ _ _

512 = 2x 2 x 2 = 256 x = 16 or

x = -16

Since the width is not a negative number, x = 16. The area of the base is 2(16), which is 32 cm 3. Module 21

IN2_MNLESE389847_U9M21L2.indd 1143

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Lesson 21.2

1143

Lesson 2

4/19/14 8:15 AM

Lesson Performance Task

QUESTIONING STRATEGIES

Genna is making a puzzle using a wooden cube. She’s going to cut the cube into three pieces. The figure below shows the lines along which she plans to cut away the first piece. The result will be a piece with four triangular sides and a square side (shaded).

The following questions lead students through a different derivation of the formula for the volume of a pyramid.

1. Each cut Genna makes will begin at the upper left corner of the cube. Write a rule describing where she drew the lines for the first piece.

Start with a cube with sides s units in length. What is its volume? V = s 3

Sample answer: Starting at the upper left corner, draw the diagonals of two adjacent sides of the cube. Then draw a diagonal through the cube from the upper left corner to the opposite corner.

Connect the 8 corners of the cube with the cube’s midpoint. Describe the shape and dimension of the figures formed. Six pyramids are formed. Each has a square base that is a face of the cube, so the base’s area is s 2. Each pyramid has a 1 height of __ s. 2

2. The figure below shows two of the lines along which Genna will cut the second piece. Draw a cube and on it, draw the two lines Genna drew. Then, using the same rule you used above, draw the third line and shade the square base of the second piece.

What is the volume of each pyramid? Explain. V = __16 s 3; 6 congruent pyramids make up s 3, the volume of the cube. Rewrite your expression for the volume of a pyramid using your expressions for the height of the pyramid, h, and the area of the base, B. 1 3 V = __ s 6

3. When Genna cut away the second piece of the puzzle, the third piece remained. Draw a new cube and then draw the lines that mark the edges of the third piece. Shade the square bottom of the third piece.

1 = __ 3

Sample answer: The volumes of the three pieces are equal because the pieces are congruent to one another.

5. Explain how the model confirms the formula for the volume of a pyramid.

INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Students can go online to find patterns for making three congruent paper pyramids that fit together to make a cube.

Sample answer: Let B represent the area of the base of the cube, h represent the height of the cube, and s represent the length of the side of the cube. Then the volume of the cube is V = s 3 = s 2s = s 2h = Bh. Each pyramid has a volume equal to 1 Bh. one-third the volume of the cube, so the volume of each pyramid is V = __ 3 Module 21

1144

2

1 = __ hB 3

© Houghton Mifflin Harcourt Publishing Company

4. Compare the volumes of the three pieces. Explain your reasoning.

(__12 s) s

Lesson 2

EXTENSION ACTIVITY IN2_MNLESE389847_U9M21L2.indd 1144

Countless dissection puzzles have been created over time, and new ones crop up every day. Students can research dissection puzzles on the Internet to find many such puzzles. Have students choose one or more puzzles that intrigue them, make copies, and challenge partners to solve them.

4/19/14 8:15 AM

Scoring Rubric 2 points: The student’s answer is an accurate and complete execution of the task or tasks. 1 point: The student’s answer contains attributes of an appropriate response but is flawed. 0 points: The student’s answer contains no attributes of an appropriate response.

Volume of Pyramids 1144

LESSON

21.3

Name

Volume of Cones

Class

Date

21.3 Volume of Cones Essential Question: How do you calculate the volumes of composite figures that include cones?

Common Core Math Standards The student is expected to: COMMON CORE

Resource Locker

G-GMD.A.1

Explore

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Also G-GMD.A.3, G-MG.A.1

Developing a Volume Formula

You can approximate the volume of a cone by finding the volumes of inscribed pyramids.

Mathematical Practices COMMON CORE

MP.4 Modeling

Language Objective

Base of inscribed pyramid has 3 sides

Explain to a partner how to apply the formulas for the volume of a cone.

A

Base of inscribed pyramid has 4 sides

The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon. Let O be the center of the cone’s base, let r be the radius of the cone, and let h be the height of the cone. Draw radii from O to the vertices of the n-gon.

ENGAGE

Break the composite figure into familiar solids, such as cones, that you have a volume formula for. Then find the volume of each figure and add them.

PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photograph. Ask students to speculate on the connection between the photo and the topic of this lesson. Then preview the Lesson Performance Task.

O r 12 r y A © Houghton Mifflin Harcourt Publishing Company

Essential Question: How do you calculate the volumes of composite figures that include cones?

Base of inscribed pyramid has 5 sides

B

x

M

_ _ Construct segment OM from O to the midpoint M of AB. How can you prove that △AOM ≅ △BOM?

¯≅ OB ¯ because they are both radii of the same circle. ¯ ¯ because M is defined OA AM ≅ BM

¯. So by the SSS Triangle Congruence as the midpoint. Both triangles share the side OM Theorem, the triangles are congruent.

B

How is ∠1 ≅ ∠2?

Since the triangles are congruent, and △AOM is a vertical reflection of △BOM, then by CPCTC, ∠1 ≅ ∠2.

Module 21

be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction

Lesson 3

1145

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Date Class Name

e of 21.3 Volum

Cones

e cones? s that includ osite figure es of comp ate the volum area of a of a circle, do you calcul ion: How circumference as for the A.1 for the formul .A.3, G-MG. al argument Also G-GMD Give an inform r, pyramid, and cone. ula G-GMD.A.1 Form cylinde a me e of circle, volum pyramids. ing a Volu inscribed Develop volumes of finding the Explore a cone by volume of imate the You can approx

Resource Locker

Quest Essential COMMON CORE

IN2_MNLESE389847_U9M21L3 1145

ed Base of inscribsides has 3 pyramid

ed Base of inscribsides has 4 pyramid

HARDCOVER PAGES 11451158 Watch for the hardcover student edition page numbers for this lesson.

ed Base of inscribsides has 5 pyramid

r n-gon. and is a regula of the cone h be the height circular base of the cone, and let ed in the id is inscrib be the radius . of a pyram base, let r The base s of the n-gon of the cone’s the vertice the center from O to Let O be Draw radii of the cone. O r 12y r B M that _ you prove . How can int M of AB ed _ the midpo M is defin from O to ¯ because ¯ ≅ BM segment OM circle. AM Construct ruence △BOM? gle Cong of the same SSS Trian △AOM ≅ both radii are ¯. So by the they OM ¯ because the side ¯ OA ≅ OB les share Both triang oint. ruent. as the midp les are cong the triang by rem, then Theo of △BOM, reflection a vertical ≅ ∠2? △AOM is How is ∠1 ruent, and are cong triangles Since the ≅ ∠2. CPCTC, ∠1

© Houghto

n Mifflin

Harcour t

Publishin

y g Compan

A

x

Lesson 3 1145 Module 21

1L3 1145

47_U9M2

ESE3898

IN2_MNL

1145

Lesson 21.3

4/19/14

8:27 AM

4/19/14 8:27 AM

C

How many triangles congruent to △AOB surround point O to make up the n-gon that is the base of the pyramid? How can this be used to find the angle measures of △AOM and △BOM?

EXPLORE

360° There are n triangles congruent to △AOB in the n-gon, so the measure of ∠AOB = ____ n , 180° ____ and the measure of ∠1 = n . Since ∠OMA ≅ ∠OMB by CPCTC, and ∠OMA and ∠OMB form

Developing a Volume Formula

a linear pair, these angles are supplementary and must have measures of 90°. So, △AOM

D

and △BOM are right triangles.

INTEGRATE TECHNOLOGY

y x , so x = rsin ∠1. In △AOM, cos ∠1 = _ In △AOM, sin ∠1 = _ r r , so y = rcos ∠1. Since ∠1 has a known value, rewrite x and y using substitution.

Students have the option of doing the Explore activity either in the book or online.

(

)

(

)

180° 180° Since x = rsin ∠1 and y = rcos ∠1, x = rsin ____ and y = rcos ____ n n .

E

QUESTIONING STRATEGIES

To write an expression for the area of the base of the pyramid, first write an expression for the area of △AOB.

To develop a formula for the volume of a cone, why does it make sense to work with inscribed pyramids? We already have a formula for the volume of a pyramid.

1 ⋅ base ⋅ height Area of △AOB = _ 2 1 ⋅ 2x ⋅ y =_ 2 = xy What is the area of △AOB, substituting the new values for x and y? What is the area of the n triangles that make up the base of the pyramid?

Area △AOB = xy

(

)

(

)

In general, what happens as the number of sides of the base of the inscribed pyramid gets larger? The volume of the pyramid gets closer to the volume of the cone.

180° 180° = rsin ____ rcos ____ n n

⎤ ⎡ 180° 180° ⎥ Area base of pyramid = n⎢rsin ____ rcos ____ n n ⎦ ⎣ ⎤ ⎡ 180° 180° ⎥ = nr 2 ⎢sin ____ cos ____ n n ⎦ ⎣

(

)

)

(

(

)

)

© Houghton Mifflin Harcourt Publishing Company

F

(

Use the area of the base of the pyramid to find an equation for the volume of the pyramid. 1 Bh Volume pyramid = _ 3 ⎡ ⎤ 180° 180° ( ) 1 ( 2) ⎢ nr sin ____ cos ____ = _ n n ⎥ h 3 ⎣ ⎦

()

( )

Module 21

(

)

1146

Lesson 3

PROFESSIONAL DEVELOPMENT IN2_MNLESE389847_U9M21L3 1146

Math Background

4/19/14 8:27 AM

The approach to finding a formula for the volume of a cone in this lesson is very similar to the approach to finding a formula for the circumference of a circle in Module 15-1, where students use inscribed regular polygons and an informal limit argument to show that the circumference, C, of a circle with radius r is given by C = 2πr. In this module, students inscribe a sequence of pyramids in a given cone and use similar reasoning to show that the volume, V, of the cone is given by 1 Bh, where B is the base area and h is the cone’s height. V = __ 3

Volume of Cones 1146

INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 When they look at models of cones, students

(

) (

)

) (

)

• View a table for the function and scroll down.

sometimes think a cone has one-half the volume of a cylinder with the same height and base. Point out that 1 the volume of the a cone has a volume equal to _ 3 associated cylinder. Encourage students to use hollow models of cones and cylinders to verify this.

What happens to the expression as n very large?

EXPLAIN 1

The value gets closer and closer to π.

Finding the Volume of a Cone

QUESTIONING STRATEGIES

(

) (

)

180° cos _ 180° gets closer to π as n becomes greater, what happens to the entire If n sin _ n n expression for the volume of the inscribed pyramid? How is the area of the circle related to the expression for the base?

1 2 πr h; πr 2 is the area of a circle. The expression gets closer to the formula for the pyramid, _ 3

Reflect

1. © Houghton Mifflin Harcourt Publishing Company

How does the volume formula for a cone compare to the volume formula for a 1 Bh , but B is pyramid? The formulas are both V = __ 3 the area of a circle, πr 2, for the cone, while B is the area of a polygon for the pyramid.

(

180° cos _ 180° . Your expression for the pyramid’s volume includes the expression n sin _ n n Use a calculator, as follows, to discover what happens to this expression as n gets larger and larger. 180° cos _ 180° as Y , using x for n. • Enter the expression n sin _ 1 n n • Go to the Table Setup menu and enter the values shown.

How is the formula for the volume of a cone related to the formula for the volume of a pyramid? 1 Bh, which Since πr 2 is the area of the base of the cone, the formula can be written as A = _ 3

is the same as the formula for the volume of a pyramid.

Explain 1

Finding the Volume of a Cone

The volume relationship for cones that you found in the Explore can be stated as the following formula.

Volume of a Cone The volume of a cone with base radius r and base area B = πr 2 and 1 2 height h is given by V = __13 Bh or by V = __ πr h. 3 You can use a formula for the volume of a cone to solve problems involving volume and capacity.

Module 21

1147

Lesson 3

COLLABORATIVE LEARNING IN2_MNLESE389847_U9M21L3 1147

Small Group Activity

4/19/14 8:27 AM

Have students each draw a cone and a cylinder, both with the same radius and height. Ask them to pass each drawing to another group member, who then measures and labels the radius and height of the cone. Have students pass the drawings again, to a group member who calculates the volume of the cone and cylinder. Have the group discuss the relationship between the volumes of the cone and the cylinder. If the volume of the cone is not exactly one-third the volume of the cylinder, have them explain why (for example, the measurements are not precise).

1147

Lesson 21.3

Example 1

The figure represents a conical paper cup. How many fluid ounces of liquid can the cup hold? Round to the nearest tenth. (Hint: 1 in 3 ≈ 0.554 fl oz.)

Find the radius and height of the cone to the nearest hundredth.

2.4 in.

AVOID COMMON ERRORS When calculating volumes of cones, some students 1 (or divide by 3). Remind may forget to multiply by __ 3 students that the volume of a cone needs the factor 1 , but the volume of a cylinder does not. Watch for of __ 3 students who calculate the volume using the slant height of the cone instead of the height.

3.9 in.

1 (2.2 in.) = 1.1 in. The radius is half of the diameter, so r = _ 2 To find the height of the cone, use the Pythagorean Theorem: r 2 + h 2 = (1.8)

2

(1.1) 2 + h 2 = (1.8) 2 1.21 + h 2 = 3.24 h 2 = 2.03, so h ≈ 1.42 in.

Find the volume of the cone in cubic inches.

(

1 π 1.1 1 πr 2h = _ V=_ 3 3

) ( 1.42 ) = 1.80 in 2

3

Find the capacity of the cone to the nearest tenth of a fluid ounce.

1.805

in 3 ≈

1.805

0.554 fl oz ≈ in 3 × _ 1 in 3

1.00

fl oz

Your Turn

Right after Cindy buys a frozen yogurt cone, her friend Maria calls her, and they talk for so long that the frozen yogurt melts before Cindy can eat it. The cone has a slant height of 3.8 in. and a diameter of 1.5 in. If the frozen yogurt has the same volume before and after melting, and when melted just fills the cone, how much frozen yogurt did Cindy before she talked to Maria, to the nearest tenth of a fluid ounce? 2.

Find the radius. Then use the Pythagorean Theorem to find the height of the cone.

r 2 + h 2 = (3.9)

2

2.4 in.

(1.2) 2 + h 2 = (3.9) 2 1.44 + h 2 = 15.21 h 2 = 13.77

3.9 in.

h ≈ 3.710 in. 3.

Find the volume of the cone in cubic inches.

2 1 1 ( V=_ πr 2h = _ π 1.2) (3.710) = 4.663 in 3 3 3

4.

© Houghton Mifflin Harcourt Publishing Company

1 (2.4 in). = 1.2 in. r=_ 2

Find the capacity of the cone to the nearest fluid ounce. 0.554 fl oz 4.63 in 3 ≈ 4.63 in 3 ×_______ ≈ 2.6 fl oz 3 1 in

Module 21

1148

Lesson 3

DIFFERENTIATE INSTRUCTION IN2_MNLESE389847_U9M21L3 1148

Modeling

4/19/14 8:27 AM

Have students make nets for a cone and a cylinder that has the same height as the cone. Then, have students cut out, fold, and tape the nets to form the threedimensional figures. Students can model the volume of the cone by filling it with dry material. Ask students to pour the material from the cone into the cylinder as many times as necessary to see how the volumes of the figures are related. Students will discover that it takes three batches of material from the cone to fill the cylinder. That is, the volume of the cone is one-third the volume of the associated cylinder.

Volume of Cones 1148

Explain 2

EXPLAIN 2

You can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure. Example 2

Finding the Volume of a Composite Figure

Find the volume of the composite figure. Round to the nearest cubic millimeter.

32 mm

INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Students should be able to recognize the solid

19 mm

16 mm

figures that make up a composite figure, including cones, and whether the volumes of those figures are to be added or subtracted from the total volume. Ask students to make an organized list of all figures comprising the composite figures and then show the formulas for their volumes. After organizing how to find the volume of the composite figure, have students find the volume of each solid, then add or subtract the volumes, as needed, to get the total.

Find the volume of the cylinder. 1 (16 mm) = 8 mm First, find the radius: r = _ 2 2 V = πr 2h = π(8) (19) = 3,820.176 … mm 3

Find the volume of the cone. The height of the cone is h = 32 mm - 19 mm = 13 mm. It has the same radius as the cylinder, r = 8 mm. 1π 1 πr 2 h = _ V=_ 3 3

( 8 ) ( 13 ) ≈ 2

871.268

mm 3

Find the total volume. Total volume = volume of cylinder + volume of cone

© Houghton Mifflin Harcourt Publishing Company

QUESTIONING STRATEGIES If a cone shares a base with a cylinder, and the volume of the cylinder is given, along with the heights of the cylinder and of the cone, how could you find the volume of the cone? Substitute the height of the cylinder and its volume into the formula for the volume of a cylinder and solve for r. Then use the value of r to find B, the area of the base of the cone. Substitute the values of B and the height into the formula for the area of a cone.

Finding the Volume of a Composite Figure

= 3,820.176 mm 3 + ≈

4,691

871.268

mm 3

mm 3

Reflect

5.

Discussion A composite figure is formed from a cone and a cylinder with the same base radius, and a . What its volume can be calculated by multiplying the volume of the cylinder by a rational number, _ b arrangements of the cylinder and cone could explain this? The heights of the cylinder and the cone must be in an integer ratio. For instance, if they have the same height, and are joined base to base, the volume of the composite figure is 1 4 4 V = πr 2h + _ πr 2h = _ πr 2h, or _ times the volume of the cylinder. 3 3 3

Module 21

1149

Lesson 3

LANGUAGE SUPPORT IN2_MNLESE389847_U9M21L3 1149

Connect Vocabulary To help students remember how to apply the formula for the volume of a cone, have students make note cards showing examples of cones with different heights and bases. Then have them use colored pencils to write the area of each base in one color, the heights in another color, and then list the formulas for the volume in different colors. Have them label each figure with the formula and volume. Invite students to share their note cards with other students.

1149

Lesson 21.3

4/19/14 8:27 AM

Your Turn

ELABORATE

Making a cone-shaped hole in the top of a cylinder forms a composite figure, so that the apex of the cone is at the base of the cylinder. Find the volume of the figure, to the nearest tenth. 6.

1( radius r = _ 3.6 cm) = 1.8 cm 2

2 V = πr 2h = π(1.8) (4.3) = 43.768 cm 3

7.

QUESTIONING STRATEGIES

4.3 cm

Find the volume of the cylinder.

How is the volume of a cone related to the volume of a cylinder with the same base and height? The volume of the cone is one-third the volume of the cylinder.

3.6 cm

Find the volume of the figure.

1 times Cone and cylinder have same height and radius, so volume of cone is _ 3

volume of cylinder. Therefore, volume of figure = volume cylinder - volume of cone

SUMMARIZE THE LESSON

1 = volume of cylinder -_ (volume of cylinder) 3

What are the main steps used to develop a formula for the volume of a cone? Sample answer: Inscribe a sequence of pyramids with ever-increasing sides in a given cone until the volume of the inscribed pyramid approaches the volume of the cone.

2 =_ (volume of cylinder) 3

2( =_ 43.768 cm 3) ≈ 29.2 cm 3 3

Elaborate 8.

Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone? Explain. Yes; the base area of the pyramid would be slightly larger than the base area of the cone,

instead of slightly smaller, but you would still be using n congruent isosceles triangles

(

)

180° with areas derived via b = r tan ____ and h = 2r. Therefore the volume of the pyramid n

(

)

180° would have a factor of n tan ____ n , and this would approach the value of π as n gets larger

9.

Essential Question Check-In How do you calculate the volumes of composite figures that include cones? Sample answer: split the figure into simpler shapes, using the volume formula of each 1 πr 2h, is comparable to the volume separate shape. The volume formula for a cone, V = _ 3 1 formula for a pyramid, V = _ Bh. 3

Module 21

IN2_MNLESE389847_U9M21L3 1150

1150

© Houghton Mifflin Harcourt Publishing Company

and larger.

Lesson 3

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Volume of Cones 1150

Evaluate: Homework and Practice

EVALUATE 1.

• Online Homework • Hints and Help • Extra Practice

Interpret the Answer Katherine is using a cone to fill a cylinder with sand. If the radii and height are equal on both objects, and Katherine fills the cone to the very top, how many cones will it take to fill the cylinder with sand? Explain your answer. It will take three cones to fill the cylinder with sand. Because the volume formula for a cylinder is V = πr 2h, and the volume formula for a cone is

ASSIGNMENT GUIDE

1 2 1 V=_ πr h, the volume of a cone is _ the volume of the cylinder. 3 3

Concepts and Skills

Practice

Explore Developing a Volume Formula

Exercise 1

Example 1 Finding the Volume of a Cone

Exercises 2–6

Example 2 Finding the Volume of a Composite Figure

Exercises 7–10

Find the volume of the cone. Round the answer to the nearest tenth. 1.9 mm

2.

6.3 ft 4.2 mm 5.9 ft 1 Bh V=_ 3

a2 + b 2 = c 2

1 πr 2h =_ 3

INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Help students determine what information

a 2 + 2.95 2 = 6.3 2

2 1 ( π 1.9) ⋅ 4.2 =_ 3

a + 8.7025 = 39.69 2

a 2 = 30.9875

1 ( π 3.61) ⋅ 4.2 =_ 3

≈ 15.9 mm 3 © Houghton Mifflin Harcourt Publishing Company

is required to use the volume formula for a cone. Sample questions: “What formula will you use?” 1 πr 2h “How can you find the radius?” Divide V = __ 3 the diameter by 2. “How can you find the height?” Draw the right triangle formed by the radius, height, and slant height and then use the Pythagorean Theorem.

3.

1 V=_ Bh 3

1 =_ π r 2h 3

2 1 ( ≈_ π 2.95) ⋅ 5.6 3

1 ( ≈_ π 8.7025) ⋅ 5.6 3

≈ 51.0 ft 3 4.

22 cm

a + 20 = 22 2

20 cm

1 Bh V=_ 3

a2 + b 2 = c 2 2

1 V=_ πr 2h 3

2

1 ( ) =_ π 84 ⋅ 20 3

a 2 + 400 = 484 a 2 = 84 cm

Module 21

Exercise

IN2_MNLESE389847_U9M21L3 1151

Lesson 21.3

≈ 1759.3 cm 3

Lesson 3

1151

Depth of Knowledge (D.O.K.)

COMMON CORE

Mathematical Practices

3 Strategic Thinking

MP.2 Reasoning

2–11

1 Recall of Information

MP.5 Using Tools

12–15

2 Skills/Concepts

MP.4 Modeling

16–19

2 Skills/Concepts

MP.5 Using Tools

20

3 Strategic Thinking

MP.2 Reasoning

21

3 Strategic Thinking

MP.3 Logic

1

1151

a ≈ 5.6 ft

4/19/14 8:27 AM

Find the volume of the cone. Leave the answer in terms of π. 5.

6.

41 m

30 in 9m

24 in.

1 V=_ Bh 3

1 V=_ Bh 3

1 =_ π r 2h 3

1 πr 2h =_ 3

1 ( )2 =_ π 9 ⋅ 41 3

1 ( )2 π 12 ⋅ 30 =_ 3 1 ( π 144) ⋅ 30 =_ 3

1 ( ) =_ π 81 ⋅ 41 3

= 1107π m 3

= 1440π in 3

Find the volume of the composite figures. Round the answer to the nearest tenth. 7.

4 in.

8 in.

8.

6 ft

6 in. 12 in.

10 ft

Volume of the large cone

Volume of cylinder

1 V=_ Bh 3 1 _ = 3 πr 2h 1 ( )2 =_ π 8 ⋅ 12

= πr 2h

= π(6) ⋅ 10 2

3

≈ 1131.0 ft 3

≈ 804.2 in 3 Volume of the small cone

Volume of cone 1 V=_ Bh 3

1 V=_ Bh 3 1 =_ π r 2h 3 1 ( )2 _ = π 4 ⋅6

1 =_ πr 2h 3

1 ( )2 =_ π 6 ⋅ 10 3

3

≈ 377.0 ft 3

≈ 100.5 in 3

1131.0 - 377.0 = 754.0

804.2 - 100.5 = 703.7 V ≈ 703.7 in

Exercise

V ≈ 754.0 ft 3

3

Module 21

IN2_MNLESE389847_U9M21L3 1152

© Houghton Mifflin Harcourt Publishing Company

V = Bh

Lesson 3

1152

Depth of Knowledge (D.O.K.)

COMMON CORE

Mathematical Practices

22

3 Strategic Thinking

MP.2 Reasoning

23

3 Strategic Thinking

MP.3 Logic

4/19/14 8:27 AM

Volume of Cones 1152

9.

AVOID COMMON ERRORS

10.

2m

When finding the volume of composite figures, some students may think they must always add the volumes of the figures that make up the composite figure. Caution students to study the composite figure carefully and then decide which volumes are added and which are subtracted. Watch for students who calculate the volume using the slant height of the cone instead of the height.

5 ft 10 ft 3 ft

13 m 12 ft

1m

Volume of cone

Volume of cone

Height of cone

1 V=_ Bh 3 1 =_ π r 2h 3 1 ( )2 _ = π 1 ⋅2

a +b =c 2

2

2

h 2 + 6 2 = 10 2 h 2 = 64

3

≈ 2.1 m 3

h = 8 ft Distance from top of cylinder to vertex of cone

Volume of cylinder V = Bh

a2 + b 2 = c 2

= πr h 2

h2 + 32 = 52

= π(1) ⋅ 11 2

h = 16 2

≈ 34.6 m 3

1 V=_ Bh 3

1 =_ πr 2h 3

1 ( )2 =_ π 6 ⋅8 3

= 96π ≈ 301.6 ft 3 Volume of cylinder V = Bh = π r 2h

= π(3) ⋅ 4 2

h = 4 ft ≈ 113.1 ft 3 Height of cylinder 301.6 - 113.1 = 188.5

2.1 + 34.6 = 36.7 V ≈ 36.7 m 3

8 - 4 = 4 ft

V ≈ 188.5 ft 3

© Houghton Mifflin Harcourt Publishing Company

11. Match the dimensions of a cone on the left with its volume on the right. B 25π ___ units 3 A. radius 3 units, height 7 units 6

B. diameter 5 units, height 2 units

D

240π units 3

C. radius 28 units, slant height 53 units

C

11,760π units 3

D. diameter 24 units, slant height 13 units.

A

21π units 3

1 1 ( )2 Volume of cone A: V = _ Bh = _ π 3 ⋅ 7 = 21π units 3 3 3 2 25 5 1 1 _ _ π units 3 Volume of cone B: V = 3 Bh = 3 π ⋅ 2= 6 2 Height of cone C: 28 2 + h 2 = 53 2, so h 2 = 2025 and h = 45

(_)

_

1 1 ( )2 Volume of cone C: V = _ Bh = _ π 28 ⋅ 45 = 11,760π units 3 3 3

Height of cone D: 12 2 + h 2 = 13 2, so h 2 = 25 and h = 5

1 1 ( )2 Volume of cone D: V = _ Bh = _ π 12 ⋅ 5 = 240π units 3 3 3

Module 21

IN2_MNLESE389847_U9M21L3 1153

1153

Lesson 21.3

1153

Lesson 3

4/19/14 8:27 AM

12. The roof of a grain silo is in the shape of a cone. The inside radius is 20 feet, and the roof is 10 feet tall. Below the cone is a cylinder 30 feet tall, with the same radius.

CONNECT VOCABULARY To remember how the volume of a cylinder is connected to the volume of a cone, have students make a graphic organizer listing the volume formulas they know and adding the volume formulas from this module.

a. What is the volume of the silo?

Volume of cone 1 V=_ Bh 3

1 ( )2 =_ π 20 ⋅ 10 3

≈ 4188.8 ft 3

Sample:

Volume of the cylinder V = Bh

Volumes of 3-Dimensional Figures

= πr 2h

= π(20) ⋅ 30

Formula

V = Bh

1 Bh V = __ 3

Figure

prism

pyramid

Figure

cylinder

cone

2

≈ 37,699.1 ft 3 4188.8 ft 3 + 37,699.1 ft 3 = 41,887.9 ft 3 V ≈ 41,887.9 ft 3 b. If one cubic foot of wheat is approximately 48 pounds, and the farmer’s crop consists of approximately 2 million pounds of wheat, will all of the wheat fit in the silo? 48 lb 41,887.9 ft 3 · = 2,010,619.2 lb 1 ft 3 Yes, the crop will fit because the silo can hold over 2 million

_

pounds of wheat. 13. A cone has a volume of 18π in 3. Which are possible dimensions of the cone? Select all that apply.

B. diameter 6 in., height 6 in. C. diameter 3 in., height 6 in. D. diameter 6 in., height 3 in. E. diameter 4 in., height 13.5 in. F. diameter 13.5 in., height 4 in.

Module 21

IN2_MNLESE389847_U9M21L3 1154

2 1 1 ( V=_ Bh = _ π 0.5) ⋅ 18 = 1.5π in 3 3 3

© Houghton Mifflin Harcourt Publishing Company • ©JenniferPhotographyImaging/iStockPhoto.com

A. diameter 1 in., height 18 in.

1 1 ( )2 V=_ Bh = _ π 3 ⋅ 6 = 18π in 3 3 3

2 1 1 ( V=_ Bh = _ π 1.5) ⋅ 6 = 4.5π in 3 3 3

1 1 ( )2 Bh = _ π 3 ⋅ 3 = 9π in 3 V=_ 3 3

1 1 ( )2 Bh = _ π 2 ⋅ 13.5 = 18π in 3 V=_ 3 3

2 1 1 ( Bh = _ π 6.75) ⋅ 4 = 91.125π in 3 V=_ 3 3

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Volume of Cones 1154

15. Roland is using a special machine to cut cones out of cylindrical pieces of wood. The machine is set to cut out two congruent cones from each piece of wood, leaving no gap in between the vertices of the cones. What is the volume of material left over after two cones are cut out?

14. The figure shows a water tank that consists of a cylinder and a cone. How many gallons of water does the tank hold? Round to the nearest gallon. (Hint: 1 ft 3 = 7.48 gal) 6 ft

12 in.

10 ft

8 ft

12 in.

Volume of cone a2 + b 2 = c 2

Volume of cylinder

a2 + 62 = 82

V = Bh

a 2 + 36 = 64 a 2 = 28

―

= π(6) ⋅ 12 2

―

≈ 1357.2 in 3

√a 2 = √28

Volume of one cone

a ≈ 5.3 cm

1 V=_ Bh 3

1 V=_ Bh 3

1 ( ) =_ π 6 ⋅6 3

1 ( )2 =_ π 6 ⋅ 5.3 3

2

≈ 226.2 in 3

≈ 200 ft 3

1357.2 - 226.2 - 226.2 = 904.8

© Houghton Mifflin Harcourt Publishing Company

Volume of cylinder = π(6) ⋅ 10 2

≈ 1131 ft 3 200 + 1131 = 1331 ft 3 1331 ft 3 ·

IN2_MNLESE389847_U9M21L3 1155

Lesson 21.3

7.48 gal ______ ≈ 9956 gal 1 ft 3

9956 gal

Module 21

1155

904.8 in 3

V = Bh

1155

Lesson 3

4/19/14 8:27 AM

16. Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x.

(

_1 Bh) 3 1 1 x - _πr h = x - _ π(_ 3 3 2) xπ -_ 12 π 1) ( _

V = s3 V = x3 = x3 = x3

2

3

2

x = x3 -

x x _1 π_

3

INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 When working with the volume of a cone

x

2

3

4

x x

formula, remind students that cones have a slant height as well as a height. The slant height is used to find the surface area of the cone, while the height of the cone is the perpendicular distance from the vertex to the base.

12

17. Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in. tall. The regular size has a 4 in. diameter. The jumbo size has an 8 in. diameter. a. Find the volume of the regular size to the nearest tenth.

8 in.

1 Bh V=_ 3

1 ( )2 =_ π 2 ⋅8 3

≈ 33.5 in 3 b. Find the volume of the jumbo size to the nearest tenth. 1 V=_ Bh 3

1 ( )2 =_ π 4 ⋅8 3

≈ 134.0 in 3 c.

The regular size costs $1.25. What would be a reasonable price for the jumbo size? Explain your reasoning.

134.0 _ =4

33.5 4 ⋅ $1.25 = $5.00 $5; the large size holds 4 times as much.

1 V=_ Bh 3

B = πr 2

1 125π = _ πr 25 3

36π = πr 2

_3 · (125π) = (_5 πr ) · _3

36 = r 2

5

6 ft = r

―

1 V=_ Bh 3

5

C = 2πr

― ― = 10π √3 C = 10π √― 3 cm = 2π(5 √3 )

= 144π V = 144π ft 3

IN2_MNLESE389847_U9M21L3 1156

2

5 √3 cm = r

1 ( =_ 36π) ⋅ 12 3

Module 21

75 = r

3

2

© Houghton Mifflin Harcourt Publishing Company

19. Find the base circumference of a cone with height 5 cm and volume 125π cm 3.

18. Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius.

1156

Lesson 3

4/19/14 8:27 AM

Volume of Cones 1156

JOURNAL

H.O.T. Focus on Higher Order Thinking

20. Analyze Relationships Popcorn is available in two cups: a square pyramid or a cone, as shown. The price of each cup of popcorn is the same. Which cup is the better deal? Explain.

Have students describe how to use the general volume formula V = Bh to write a formula for the volume of a cone with base radius r and height h. Use figures to illustrate the steps.

12 cm

12 cm

The pyramid is the better deal because you get a 20 cm

greater volume of popcorn (960 cm 3 versus about 3

754 cm ) for the same price. Volume of pyramid

Volume of cone

1 V=_ Bh 3

1 V=_ Bh 3

1 ( 2) =_ 12 ⋅ 20

1 ( ) =_ π 6 ⋅ 20 3 2

3

= 960 cm 3

≈ 754.0 in 3

21. Make a Conjecture A cylinder has a radius of 5 in. and a height of 3 in. Without calculating the volumes, find the height of a cone with the same base and the same volume as the cylinder. Explain your reasoning.

h = 9 in.; the volume of a cone with the same base and height as the 1 the volume of the cylinder. For the cone to have the same cylinder is __ 3

volume as the cylinder, the height of the cone must be 3 times the height of the cylinder.

© Houghton Mifflin Harcourt Publishing Company

22. Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinder’s base, as shown. Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood. 2 The solid has __ the volume of the cylinder since the cone 3 1 that is removed has __ the volume of the cylinder. 3

23. Explain the Error Which volume is incorrect? Explain the error.

A

1 ( 8 2π)( 17) V = __ 3 1088π 3 = _______ cm 3

B

1 ( 8 2π)( 15) V = __ 3 = 320π cm

15 cm 17 cm

3

8 cm

The calculation show in A is incorrect because it uses the slant height of the cone instead of the height.

Module 21

IN2_MNLESE389847_U9M21L3 1157

1157

Lesson 21.3

1157

Lesson 3

4/19/14 8:27 AM

Lesson Performance Task

INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Compare and contrast pyramids and cones.

You’ve just set up your tent on the first night of a camping trip that you’ve been looking forward to for a long time. Unfortunately, mosquitoes have been looking forward to your arrival even more than you have. When you turn on your flashlight you see swarms of them—an average of 800 mosquitos per square meter, in fact.

How are they alike? How are they different? Sample answer: Both are three-dimensional shapes that taper from a plane figure to a point. The plane figure is a polygon for a pyramid, a circle for a cone. A pyramid has triangles for lateral sides, a cone does not. The volume of both is found the same way: Find 1 , the area of its base, and its height. the product of __ 3

Since you’re always looking for a way to use geometry, you decide to solve a problem: How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram), and how many are in the second three meters (Zone 2)? Zone 2 Zone 1 30° 30°

3m

3m

1. Explain how you can find the volume of the Zone 1 cone.

INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 A surface of rotation is generated by revolving

2. Find the volume of the Zone 1 cone. Write your answer in terms of π. 3. Explain how you can find the volume of the Zone 2 cone. 4. Find the volume of the Zone 2 cone. Write your answer in terms of π. 5. How many more mosquitoes are there in Zone 2 than there are in Zone 1? Use 3.14 for π. 1. Sample answer: Find the radius of the circular base of the Zone 1 cone. Then use A = π r 2 to find B, the area of the base of the cone. Finally, use V = 1 Bh, with 3 h = 3 meters, to find the volume of the cone.

a shape about a line called the axis of rotation. For example, if you rotate a half circle about a line that is a diameter of the full circle (the original circle), you generate a sphere. Describe how, using a shape and an axis of rotation, you could generate a cone. Sample answer: Use a right triangle as your shape. Use a line that contains one of the triangle’s legs as the axis of rotation.

_

2.

The radius of the base of the Zone 1 cone is the length of the side opposite the 30° 3 angle in a 30°-60°-90° triangle with a medium side measuring 3 m: r = = √3 √3 2 2 √ ) ( B = πr = π 3 = 3π 1 V = (3π) 3 = 3π cubic meters 3 Sample answer: Find the volume of the combined Zone 1/Zone 2 cone, using the above

3.

―

―

method. Subtract from that volume the volume of the Zone 1 cone. 4.

The radius of the base of the Zone 1/Zone 2 cone is the length of the side opposite the 6 30° angle in a 30°-60°-90° triangle with a medium side measuring 6 m: r = = 2 √3 √ 3 2 2 B = π r = π (2 √3 ) = 12π 1 V = (12π) 6 = 24π cubic meters 3 Volume of Zone 2 = 24π - 3π = 21π cubic meters

_

5.

_ ―

―

―

© Houghton Mifflin Harcourt Publishing Company

_

_ ―

Number in Zone 1 = 3π(800) = 2400(3.14) = 7536

Number in Zone 2 = 21π(800) = 16,800(3.14) = 52,752

There are 45,216 more mosquitos in Zone 2 than there are in Zone 1. Module 21

1158

Lesson 3

EXTENSION ACTIVITY IN2_MNLESE389847_U9M21L3 1158

Ask students to complete the sentences below.

4/19/14 8:27 AM

1. If you double the radius of the base of a cone, the volume of the cone is ____. multiplied by 4 2. If you double the height of a cone, the volume of the cone is ____. multiplied by 2 3. If you double both the height of a cone and the radius of its base, the volume of the cone is ____. multiplied by 8

Scoring Rubric 2 points: The student’s answer is an accurate and complete execution of the task or tasks. 1 point: The student’s answer contains attributes of an appropriate response but is flawed. 0 points: The student’s answer contains no attributes of an appropriate response.

Volume of Cones 1158

LESSON

21.4

Name

Volume of Spheres

Class

Date

21.4 Volume of Spheres Essential Question: How can you use the formula for the volume of a sphere to calculate the volumes of composite figures?

Common Core Math Standards The student is expected to: COMMON CORE

Resource Locker

G-GMD.A.2(+)

Explore

Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Also G-GMD.A.3, G-MG.A.1, G-MG.A.2

Developing a Volume Formula

To find the volume of a sphere, compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed.

Mathematical Practices COMMON CORE

r

r

MP.2 Reasoning

r

Language Objective Explain to a partner how to apply the formula for the volume of a sphere.

A

ENGAGE

View the Engage section online. Discuss the photograph. Ask students to identify the subject of the photo and to judge whether or not it is built to scale. Then preview the Lesson Performance Task.

x r

―――

By the Pythagorean Theorem, x 2 + R 2 = r 2. Solving for R produces R = √r 2 + x 2 . To find the area of the disk that is a cross-section, use the area for a circle, A = πR 2, or _

© Houghton Mifflin Harcourt Publishing Company

PREVIEW: LESSON PERFORMANCE TASK

The region of a plane that intersects a solid figure is called a cross section. To show that cross sections have the same level at every base, use the Pythagorean Theorem to find a relationship between r, x, and R. R

Essential Question: How can you use the formula for the volume of a sphere to calculate the volumes of composite figures? Break the figures into familiar figures, including spheres and hemispheres, for which you know the volume formulas. Then find the individual volumes and add them.

r

A = π(√ r 2 - x 2 ) . So A disk = π(r 2 - x 2).

B

2

A cross section of the cylinder with the cone removed is a ring.

r

To find the area of the inner ring, find the area of the outer circle and of the inner circle. Then subtract the area of the inner circle from the outer circle.

x x r

The outer circle of the ring has a radius of r, so its area is A outer = πr 2. The inner circle has radius of x, so its area is A inner = πx 2. The area of ring-shaped cross section is: A ring = A outer-A inner = πr 2 - πx 2

= π(r 2-x 2)

The cross-sectional areas of the disk and the ring are equal.

Module 21

be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction

Lesson 4

1159

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Date Class Name

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Resource Locker

Quest Essential

HARDCOVER PAGES 11591170

COMMON CORE

IN2_MNLESE389847_U9M21L4 1159

r

Watch for the hardcover student edition page numbers for this lesson.

r

r n. To show a cross sectio em to figure is called Pythagorean Theor cts a solid the base, use that interse of a plane level at every the same The region R. sections have en r, x, and that cross R nship betwe find a relatio

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x

r

―――

2 2 +x . R = √r 2 or produces , ng for R 2 , A = πR 2 R2 = r . Solvi for a circle rem, x + use the area gorean Theo -section, is a cross By the Pytha disk that area of the 2 x2 ). To find the r = π(r _2 2 A disk So ) . √ 2-x A = π( r is a ring. x removed cone the the of er with circle and x n of the cylind of the outer circle. r A cross sectio find the area circle from the outer inner ring, of the inner area of the ct the area To find the Then subtra 2 r, so its area of s . inner circle. x has a radiu A inner = π the ring its area is circle of s of x, so The outer has radiu -A inner 2 inner circle A ring = A outer πr . The 2 is A outer = section is: 2 cross x d π hape = πr of ring-s The area 2 x2 ) = π(r equal. ring are and the the disk areas of -sectional The cross

© Houghto

n Mifflin

Harcour t

Publishin

Lesson 4

1159 Module 21

1L4 1159

47_U9M2

ESE3898

IN2_MNL

1159

Lesson 21.4

4/19/14

8:35 AM

4/19/14 8:32 AM

C

Find an expression for the volume of the cylinder with the cone removed. The volume of the cylinder is: V cylinder = πr 2h

EXPLORE

= π r 2r = πr

3

Developing a Volume Formula

1 2 The volume of the cone is: V cone = _ πr h 3 1 2 =_ πr r 3 1 3 =_ πr 3

INTEGRATE TECHNOLOGY

1 3 The volume of cylinder with the cone removed is: V cylinder − V cone = _ πr −πr 3 3 2 3 =_ πr 3

D

Students have the option of doing the Explore activity either in the book or online.

Use Cavalieri’s principle to deduce the volume of a sphere with radius r.

The sphere and cylinder with the cone removed have the same height and the same cross-sectional area at every level x. By Cavalieri’s principle, the two figures have the same

QUESTIONING STRATEGIES

2 3 volume. Since the volume of the hemisphere is _ πr , a sphere with radius r has twice this 3

4 3 volume, or V sphere = _ πr . 3

What do you need to show to use Cavalieri’s Principle? The figures have the same cross-sectional area at every level.

Reflect

1.

How do you know that the height h of the cylinder with the cone removed is equal to the radius r? Possible answer: The height of the cylinder is the same as the height of the hemisphere,

What does Cavalieri’s Principle allow you to conclude? The two figures have the same volume.

which must be r. 2.

What happens to the cross-sectional areas when x = 0? when x = r? Possible answer: When x = 0, the cross section of the hemisphere is a circle with

INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Review the definition of sphere and discuss

the same radius r, and the ring-shaped cross section is also a circle with radius r (with a point of zero area removed), so both have area πr 2 . When x = 0, the cross section © Houghton Mifflin Harcourt Publishing Company

of the hemisphere is a point with zero area, and the ring-shaped cross section is a ring with radius r but zero width, so also has zero area.

Module 21

1160

the related terms hemisphere and great circle. Explain to students that when a plane intersects a sphere, the cross-section that is formed is either a single point or a circle. If the plane passes through the center of the sphere, the cross section is a great circle.

Lesson 4

PROFESSIONAL DEVELOPMENT IN2_MNLESE389847_U9M21L4 1160

Integrate Mathematical Practices

4/19/14 8:32 AM

This lesson provides an opportunity to address Mathematical Practice MP.2, which calls for students to “reason abstractly and quantitatively.” Students are already familiar with Cavalieri’s Principle but, in this module, a surprising application of this principle is used. The argument is based on showing that a hemisphere and a cylinder from which a cone has been removed have the same cross-sectional area at every level and therefore must have the same volume. A bit 4 πr 3. of algebra shows that the volume of a sphere is equal to __ 3

Volume of Spheres 1160

Explain 1

EXPLAIN 1

Finding the Volume of a Sphere

The relationship you discovered in the Explore can be state as a volume formula.

Volume of a Sphere

Finding the Volume of a Sphere

4 πr 3. The volume of a sphere with radius r is given by V = _ 3 You can use a formula for the volume of a sphere to solve problems involving volume and capacity.

QUESTIONING STRATEGIES

Example 1

What dimension or dimensions do you need to know to find the volume of a sphere? the radius

The figure represents a spherical helium-filled balloon. This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon, as it cruises at an altitude of 500 ft. How much helium, to the nearest hundred gallons, does the balloon hold? Round to the nearest tenth. (Hint: 1 gal ≈ 0.1337 ft 3)

72 ft

Step 1 Find the radius of the balloon. 1 (72 ft) = 36 ft The radius is half of the diameter, so r = _ 2 Step 2 Find the volume of the balloon in cubic feet.

AVOID COMMON ERRORS When calculating volumes of spheres, some students 4 instead of __ 1 , as in the may forget to multiply by __ 3 3 previous volume formulas. Remind students that the 4 . Watch for volume of a sphere needs the factor of __ 3 students who calculate the volume using the second power of the radius instead of the third power.

4 πr 3 V=_ 3

( )

3

4 π 36 =_ 3 ≈

195,432.195 ft 3

Step 3 Find the capacity of the balloon to the nearest gallon. ft 3 ≈

195,432.195

195,432.195

1 gal ft 3× _3 ≈ 0.1337 ft

1,462,000

gal

Your Turn

© Houghton Mifflin Harcourt Publishing Company

A spherical water tank has a diameter of 27 m. How much water can the tank hold, to the nearest liter? (Hint: 1,000 L = 1 m3) 3.

Find the volume of the tank in cubic meters. 1( 27 m) = 13.5 m r=_ 2

4 3 _ V=_ πr = 43π(13.5 m) = 10,305.9947... m 3 3 3

4.

Find the capacity of the tank to the nearest liter.

10,305.9947... m 3 ≈ 10,305.9947... m 3 × _____ ≈ 10,305,995 L 3 1,000 L 1m

Module 21

1161

Lesson 4

COLLABORATIVE LEARNING IN2_MNLESE389847_U9M21L4 1161

Whole Class Activity Have students each draw a sphere and label the radius or the diameter. Then have each student pass the drawing to another class member who then calculates the volume of the sphere. Discuss with the class the relationship between the volumes of the spheres, given the diameter or the radius. Ask them how the volume would change if the radius is doubled or tripled. Also ask how they would find the volume of a hemisphere.

1161

Lesson 21.4

4/19/14 8:32 AM

Finding the Volume of a Composite Figure

Explain 2

EXPLAIN 2

You can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure. Example 2

Find the volume of the composite figure. Round to the nearest cubic centimeter.

5 cm

Finding the Volume of a Composite Figure

13 cm

Step 1 Find the volume of the hemisphere. 2 πr 3 = _ 2 π(5) 3 ≈ 261.799 cm 3 V=_ 3 3 Step 2 Find the height of the cone.

(

h2 +

) ( 2

5

=

13

h 2 + 25 = 169

)

INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Students should be able to recognize the solid

Step 3 Find the volume of the cone.

2

The cone has the same radius as the hemisphere, r = 5 cm.

h 2 = 144

1 πr 2h V=_ 3

h = 12

(

1π =_ 3

figures that make up a composite figure, including spheres, and to know whether the volumes of those figures are to be added or subtracted from the total volume. Have students make an organized list of the volume formulas needed to find the volume of the composite figure. Then have them find the volume of each solid, and add or subtract the volumes, as needed, to get the total.

) ( 12 ) 2

5

= 314.159 cm 3 Step 4 Find the total volume. Total volume = volume of hemisphere + volume of cone =

261.799

cm 3 +

314.159

cm 3

≈ 576 cm 3 Reflect

5.

than its radius. Your Turn

A composite figure is a cylinder with a hemispherical hole in the top. The bottom of the hemisphere is tangent to the base of the cylinder. Find the volume of the figure, to the nearest tenth.

7 in.

Volume of cylinder:

1( 7 in.) = 3.5 in; height h = r = 3.5 in. radius r = _ 2

How do you calculate the volume of a composite figure that includes a sphere? Separate the figure into separate solids for which known volume formulas apply. Then add or subtract the volumes of each separate solid, depending on how the composite figure is formed.

© Houghton Mifflin Harcourt Publishing Company

part of it would have to lie outside the cone, even if the cone’s height were much greater

6.

QUESTIONING STRATEGIES

Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius? No; possible answer: at the widest part of the hemisphere, its surface is almost vertical, so

AVOID COMMON ERRORS

2 V = πr 2 h = π(3.5) (3.5) = 134.695 in 3

3 2 3 _ Volume of hemisphere: V = _ πr = 2 π(3.5) = 89.797 in 3

3

3

Volume of figure = volume of cylinder - volume of hemisphere = 134.695 in 3 - 89.797 in 3 ≈ 44.9 in 3

Module 21

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Lesson 4

DIFFERENTIATE INSTRUCTION IN2_MNLESE389847_U9M21L4 1162

Kinesthetic Experience

4/19/14 8:32 AM

Some students may have difficulty remembering to use the correct units of measure in their answers. Encourage students to list the measurements of spheres that are given in linear, square, and cubic units. Linear units are used for a radius, a diameter, or a circumference; square units are used for area or surface area (to be studied later); and cubic units are used for volume.

Have groups of students brainstorm about how to find the radius of a real-world sphere (for example, use a measuring tape to find the circumference and then calculate the radius from the circumference formula). Then have them find the volumes of the types of spherical balls used in sports. Have them make a poster listing the sport, the size of the ball, and its volume. Invite students to share their posters with the class.

Volume of Spheres 1162

Elaborate

ELABORATE

7.

QUESTIONING STRATEGIES

Discussion Could you use an inscribed prism to derive the volume of a hemisphere? Why or why not? Are there any other ways you could approximate a hemisphere, and what problems would you encounter in finding its volume? Possible answer: no; no matter how many sides the base of the prism has, it will always be significantly smaller than the hemisphere; you could use discs (very thin cylinders) to

How do you find the volume of a sphere given the diameter? The radius is one-half the 4 π times the diameter. The volume of the sphere is __ 3 cube of the radius.

approximate the hemisphere, but you would need some way to add up their volumes. 8.

Essential Question Check-In A gumball is in the shape of a sphere, with a spherical hole in the center. How might you calculate the volume of the gumball? What measurements are needed?

I could subtract the volume of the spherical hole from the volume of the gumball. I would need to know the radius of both the gumball and the hole.

SUMMARIZE THE LESSON Have students make a graphic organizer summarizing the volume formulas they have learned in this module. Sample:

Volume Formula

Prism

V = Bh

Cylinder

V = Bh 1 Bh V = __ 3 1 Bh V = __ 3 4 πr 3 V = __ 3

Pyramid Cone Sphere

Evaluate: Homework and Practice 1.

© Houghton Mifflin Harcourt Publishing Company

ThreeDimensional Figure

Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder.

• Online Homework • Hints and Help • Extra Practice

Pick an arbitrary radius common to the sphere and the cylinder. Let r = 1 unit, which would make the height of the cylinder 2 units V cylinder = Bh

= π(1 2)(2) = 2π

4 3 V sphere = _ πr 3

4 ( ) =_ π 1 3

3

4π = __ 3 3 3 4π __ 2π ÷ = 2π ∙ __ =_ = 1.5 3

4π

2

The volume of the cylinder is 1.5 times the volume of the sphere.

Module 21

1163

Lesson 4

LANGUAGE SUPPORT IN2_MNLESE389847_U9M21L4 1163

Connect Vocabulary To help students remember how to apply the formula for the volume of a sphere, have students make a small poster showing examples of spheres with different diameters or radii. Then have them use colored pencils to write the radius in one color and the formulas for the volume in a different color. Have them label each figure with its formula and volume. Invite students to share their posters with the class.

1163

Lesson 21.4

4/19/14 8:32 AM

Find the volume of the sphere. Round the answer to the nearest tenth. 2.

3.

3.7 in.

EVALUATE

4. 11 ft

4 3 V=_ πr 3

4 ( =_ π 3.7) 3

Circumfernce of great circle is 14π cm

4 3 πr V=_ 3

4 ( =_ π 5.5) 3

3

≈ 212.2

C = 2 πr 14π = 2πr

3

V = 212.2 in 3

ASSIGNMENT GUIDE

7 cm = r

≈ 696.9

4 3 V=_ πr 3

V = 696.9 ft 3

4 ( ) =_ π 7 3

3

≈ 1436.8 V = 1436.8 cm 3 Find the volume of the sphere. Leave the answer in terms of π. 5.

6.

7. 1m

4 3 V=_ πr 3

_4 πr 3 4 _ = π(1) 3 4π _ = 3 4π V= _m

4 ( )3 =_ π 10 3

_ = 1333.3π _ V = 1333.3π cm 3

V=

A = πr 2

3

3

_ _

3

9 in. = r

3

4 3 V=_ πr 3

4 ( ) =_ π 9 3

3

≈ 972π V = 972π in 3

Module 21

Exercise

IN2_MNLESE389847_U9M21L4 1164

COMMON CORE

Mathematical Practices

3 Strategic Thinking

MP.6 Precision

2–11

1 Recall of Information

MP.5 Using Tools

12–16

2 Skills/Concepts

MP.4 Modeling

17–20

2 Skills/Concepts

MP.5 Using Tools

21

3 Strategic Thinking

MP.6 Precision

22

3 Strategic Thinking

MP.2 Reasoning

23

3 Strategic Thinking

MP.1 Problem Solving

1

Explore Developing a Volume Formula

Exercise 1

Example 1 Finding the Volume of a Sphere

Exercises 2–7

Example 2 Finding the Volume of a Composite Figure

Exercises 8–14

required to use the volume formula for a sphere. Sample questions: “What formula will you use?” 4 V = πr 3. “How can you find the radius?” Divide 3 the diameter by 2. “How can you find the volume of a hemisphere?” Find half the volume of the related sphere.

_

Lesson 4

1164

Depth of Knowledge (D.O.K.)

Practice

INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Help students determine what information is

81π = πr 2 81π πr 2 π = π 81 = r 2

© Houghton Mifflin Harcourt Publishing Company

20 cm

A = 81π in2

Concepts and Skills

4/19/14 8:32 AM

Volume of Spheres 1164

AVOID COMMON ERRORS

Find the volume of the composite figure. Leave the answer in terms of π.

For hemisphere problems, watch for students who find the volume of the entire sphere instead of half 2 πr 3 can the sphere. Point out that the formula V = __ 3 be used for the volume of a hemisphere.

8.

5 ft

9.

2 ft

2 in.

8 in.

3 in.

V cylinder = Bh

V cylinder = Bh

= π∙2 ∙ 5

= π∙8 2 ∙ 3

= 20π ft 3

= 192π in 3

2

(

2 ( 3) =_ π 2 3 16π 3 = ___ in

to make a sphere) 4 ( ) =_ π 2 3

)

1 _ 4 3 V hemisphere = _ πr 2 3

4 3 V sphere = _ πr (two hemispheres combine 3 3

32π 3 = ___ ft 3

32π 60π 32π 92π 32π 3 20π + ___ =_ ∙ 20π + ___ = ___ + ___ = ___ 3 3 3 3 3 3 92π 3 V total = ___ f t 3

V total

3

16π _3 ∙ 192π - _ 3 3 16π 576π =_-_ 3 3 560π =_ 3 560π = _ in

16π 192π - ___ = 3

3

3

Find the volume of the composite figure. Round the answer to the nearest tenth. 10.

11.

3 cm

10 mm

4 cm © Houghton Mifflin Harcourt Publishing Company

8 mm

5 cm

10 cm

1 V cone = _ Bh 3

V prism = Bh

1 =_ π ∙ 10 2 ∙ 24 3

= 10 ∙ 5 ∙ 4 = 200 cm

(

≈ 2513.3 mm 3 1 4 πr 3 V hemisphere = _ 2 3 2 ( 3) =_ π 8 3

3

(_ )

)

1 _ 4 3 V hemisphere = _ πr 2 3 2 ( =_ π 3 3

24 mm

3

)

= 18π

≈ 1072.3 mm 3

≈ 56.5 cm 3

2513.3 - 1072.3 = 1441

200 + 56.5 = 256.5

V total ≈ 1441 mm 3

V total ≈ 256.5

Module 21

IN2_MNLESE389847_U9M21L4 1165

1165

Lesson 21.4

1165

Lesson 4

4/19/14 8:32 AM

12. Analyze Relationships Approximately how many times the volume of a grapefruit with diameter 10 cm is the volume of a lime with diameter 5 cm?

(

)

1 _ 4 3 V grapefruit = _ πr 2 3

2 ( 3) =_ π 10 3

2000 π = _____ 3

( 3 ) ____ _____ = 2000 = 8 2000π _____

250π (____ 3 )

(

PEERTOPEER DISCUSSION Have students work in pairs. Each student should make up two problems, one involving the volume of a sphere and one involving the volume of a composite figure that includes a sphere. Then have students exchange problems with their partners, and each explain to the other how to solve the problem that the partner wrote.

)

1 _ 4 3 πr V lime = _ 2 3 2 ( 3) =_ π 5 3 250π = ____ 3

250

The volume of the grapefruit is about 8 times the volume of the lime. 13. A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter. Estimate the volume of the bead to the nearest whole. 4 πr 3 V sphere = _ 3

4 ( 3) =_ π 4 3

≈ 268 mm 3

V cylinder = Bh

= π(1 2)(8)

= 8π ≈ 25 mm

3

V bead = V sphere - V cylinder ≈ 268 - 25 = 243 mm 3 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Jasmina81/iStockPhoto.com

14. Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r.

(

)

1 _ 4 3 πr + Bh V=_ 2 3

3 2 3 πr ) + (2r) = (_ 3 2 3 =_ πr + 8r 3 3 2 = r 3(_ π + 8) 3 2 2 3 3 _ r ( π + 8) or _ πr + 8r 3

3

Module 21

IN2_MNLESE389847_U9M21L4 1166

3

1166

Lesson 4

4/19/14 8:32 AM

Volume of Spheres 1166

15. One gallon of propane yields approximately 91,500 BTU. About how many BTUs does the spherical storage tankshown provide? Round to the nearest billion BTUs. (Hint: 1 ft 3 ≈ 7.48 gal)

5 ft

4 3 V=_ πr 3

4 ( ) =_ π 5 3

3

≈ 523.6 ft3

523.6 ft 3 ∙ ______ ≈ 3916.5 gal 3 7.48 gal 1 ft

3916.5 gal ∙ ________ ≈ 358,000,000 BTU 91,500 BTU 1 gal

16. The aquarium shown is a rectangular prism that is filled with water. You drop a spherical ball with a diameter of 6 inches into the aquarium. The ball sinks, causing the water to spill from the tank. How much water is left in the tank? Express your answer to the nearest tenth. (Hint: 1 in. 3 ≈ 0.00433 gal)

12 in.

4 πr 3 V sphere = _ 3

V prism = Bh

20 in.

4 ( 3) =_ π 3 3

= (20 ∙ 12)(12)

= 2880 in 3

12 in.

≈ 113.1 in 3

V prism - V sphere = 2800 - 113.1 ≈ 2686.9 in 3 2686.9 in 3 ∙ ________ ≈ 11.6 gal 3 0.00433 gal 1 in

The amount of water left is about 8.6 gallons.

© Houghton Mifflin Harcourt Publishing Company

17. A sphere with diameter 8 cm is inscribed in a cube. Find the ratio of the volume of the cube to the volume of the sphere. 6 4 A. _ V cube = Bh πr 3 V sphere = _ π 3 2 B. _ 3π 3π C. _ 4 3π D. _ 2

Module 21

IN2_MNLESE389847_U9M21L4 1167

1167

Lesson 21.4

= 83

= 512 cm

=

3

_4 π(4 ) 3

3

256 π in 3 = ___ 3

(3 ∙ 512) 512 1536 6 _____ = _______ = ____ = __ 256 π) (___ 3

(

256 3 ∙ ___ π 3

)

1167

256π

π

Lesson 4

4/19/14 8:32 AM

For Exercises 18–20, use the table. Round each volume to the nearest billion π.

INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Point out that if the radius of a sphere is

Diameter (mi)

Planet Mercury

3,032

Venus

7,521

Earth

7,926

Mars

4,222

Jupiter

88,846

Saturn

74,898

Uranus

31,763

Neptune

30,775

doubled, then the volume of the sphere is multiplied by the cube of the dimension change, or 8. If the dimensions of the sphere change by a factor of k, then the volumes are in the ratio 1:k 3.

18. Explain the Error Margaret used the mathematics shown to find the volume of Saturn. 4 π 74,898 2 ≈ _ 4 πr 2 = _ V=_ ( ) 4 π(6,000,000,000) ≈ 8,000,000,000π 3 3 3 Explain the two errors Margaret made, then give the correct answer. Margaret used the diameter rather than the radius, which she squared rather than cubed. 4 πr 3 = _ 4 π 37,449 3 ≈ _ V=_ ( ) 4 π(52,520,000,000,000) ≈ 70,027,000,000,000π 3 3 3 The correct answer is 70,027,000,000,000π. 19. The sum of the volumes of Venus and Mars is about equal to the volume of which planet?

Volume of Venus 4 π(3760.5)3 V=_ 3 ≈ 71,000,000,000π

Volume of Mars 4 π(2111)3 V=_ 3 ≈ 12,000,000,000π

71,000,000,000π + 12,000,000,000π = 83,000,000,000π © Houghton Mifflin Harcourt Publishing Company

Volume of Earth 4 π(3963) 3 V=_ 3 ≈ 83,000,000,000π Volume of Venus + Volume of Mars = Volume of Earth 20. How many times as great as the volume of the smallest planet is the volume of the largest planet? Round to the nearest thousand.

Volume of Jupiter 4 π 44,423 3 V=_ ) ( 3 ≈ 116,885,000,000,000π

Volume of Mercury 4 π(1516) 3 V =_ 3 ≈ 4,000,000,000π

116,885,000,000,000π _ 116,885 ___ = 4,000,000,000π

4 ≈ 29,200

About 29,000 times as great.

Module 21

IN2_MNLESE389847_U9M21L4 1168

1168

Lesson 4

4/19/14 8:31 AM

Volume of Spheres 1168

H.O.T. Focus on Higher Order Thinking

JOURNAL

21. Make a Conjecture The bathysphere was an early version of a submarine, invented in the 1930s. The inside diameter of the bathysphere was 54 inches, and the steel used to make the sphere was 1.5 inches thick. It had three 8-inch diameter windows. Estimate the volume of steel used to make the bathysphere. Possible solution: 4 4 V bathysphere = πr 3 V inside = πr 3 V window = Bh 3 3 3 3 4 ( 4 = π(4 2)(1.5) = π 28.5) = π(27) 3 3 ≈ 75 in 3 ≈ 96,967 in 3 ≈ 82,448 in 3

Have students describe how they would determine which solid has the greater volume: a cube with side length 2r or a sphere with diameter 2r.

_ _

_ _

Amount of steel used: 96,967 - 82,448 - 3(75) = 96,967 - 82,448 - 225 = 14,294 in 3 22. Explain the Error A student solved the problem shown. Explain the student’s error and give the correct answer to the problem. A spherical gasoline tank has a radius of 0.5 ft. When filled, the tank provides 446,483 BTU. How many BTUs does one gallon of gasoline yield? Round to the nearest 3 thousand BTUs and use the fact that 1 ft ≈ 7.48 gal. 3 The volume of the tank is __43 πr 3 = __43 π(0.5) ft 3. Multiplying by 7.48 shows that this is approximately 3.92 gal. So the number of BTUs in one gallon of gasoline is approximately 446,483 × 3.92 ≈ 1,750,000 BTU.

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Regien Paassen/Shutterstock

The student should have divided the total number of BTUs by 3.92 instead of multiplying; the correct answer is 114,000 BTU. 23. Persevere in Problem Solving The top of a gumball machine is an 18 in. sphere. The machine holds a maximum of 3300 gumballs, which leaves about 43% of the space in the machine empty. Estimate the diameter of each gumball. 3 4 4 V sphere = πr 3 = π(9) ≈ 3054 in 3 3 3 3054 ∙ (1 - 0.43) = 3054 ∙ 0.57 = 1740.78 in 3 1740.78 ≈ 0.5275 3300 4 3 0.5275 = πr 3 0.3956 ≈ πr 3

_

_

_

_

0.1259 ≈ r 3 ――― 3 ― √0.1259 ≈ √r 3 3

0.5 in. ≈ r

The diameter is twice the radius, so the diameter = 2(0.5) = 1. The diameter of one gumball is approximately 1 in.

Module 21

IN2_MNLESE389847_U9M21L4 1169

1169

Lesson 21.4

1169

Lesson 4

4/19/14 8:31 AM

Lesson Performance Task

AVOID COMMON ERRORS Unit conversions with huge numbers, such as are encountered in the Lesson Performance Task, can be simplified through the careful use of scientific notation and unit conversions: 9.3 × 10 7 miles = _____________ 2 × 10 3 miles ____________ inch 9.3 × 10 7 - 3 miles⋅_____ inch = 4.65 × 10 4 inches ___ 2 miles

For his science project, Bizbo has decided to build a scale model of the solar system. He starts with a grapefruit with a radius of 2 inches to represent Earth. His “Earth” weighs 0.5 pounds. Find each of the following for Bizbo’s model. Use the rounded figures in the table. Round your answers to two significant figures. Use 3.14 for π.

1. the scale of Bizbo’s model: 1 inch =

miles

( )

2. Earth’s distance from the Sun, in inches and in miles 3. Neptune’s distance from the Sun, in inches and in miles 4. the Sun’s volume, in cubic inches and cubic feet

Note: Answers may differ slightly due 5. the Sun’s weight, in pounds and in tons (Note: the Sun’s density is 0.26 times the Earth’s density.) to rounding variations. 7 9.3 × 10 miles 1. 2 inches = 4000 miles 2. = 46,500 inches 2000 miles ________ inch

46,500 inches ≈ 47,000 inches

46,500 inches __ ≈ 0.73 miles 3.

creation of the light year, a unit that makes dealing with such distances far easier. There is no commonly used unit to measure the huge volumes of the stars, however, which are often given in cubic miles. Invent a unit that allows the volumes of stars to be expressed as manageable numbers, such as 10 or 100. Use your unit to express the volume of a star other than the Sun. Sample answer: S, the volume of the Sun

63,360 inches __________

2.8 × 10 miles __ = 1,400,000 inches

1 mile

9

2000 miles ________ inch

1,400,000 inches __ ≈ 22 miles 63,360 inches __________ 1 mile

4. The radius of Bizbo’s model sun is

4.3 × 10 miles __ = 215 inches. 5

2000 miles ________ inch

_4 πr = _4 (3.14)(215) 3

3

4.2 × 10 cubic inches ___ ≈ 24,000 cubic feet

3

≈ 4.2 × 10 cubic inches.

3

≈ 33.5 cubic inches

3

7

7

1728 cubic inches _____________ 1 cubic foot

5. First compute the volume of the model Earth: V =

_4 (3.14)(2) 3

Now find the density of the model Earth: 0.5 pounds ≈ 0.0149 pounds per cubic inch D= 33.5 cubic inches The sun’s density is 0.26 times the density of the model Earth:

__

D = (0.26)(0.0149) ≈ 0.003874 pounds per cubic inch

© Houghton Mifflin Harcourt Publishing Company

The volume of Bizbo’s model sun is V =

)

INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 The huge distances to the stars led to the

__

1 inch = 2000 miles

(

Multiply this by the sun’s volume to find its weight:

(0.003874 pounds per cubic inch)(4.2 × 10 7 cubis inches) ≈ 160,000 pounds 160,000 pounds __ = 80 tons 2000 pounds __________ 1 ton

Module 21

1170

Lesson 4

EXTENSION ACTIVITY IN2_MNLESE389847_U9M21L4 1170

Give these directions:

4/19/14 8:31 AM

1. Research the meaning and calculate the length of a light year. distance light travels in 1 year; about 6 trillion miles 2. Calculate the mileage to the nearest star and at least 5 other stars. nearest: Proxima Centauri; 4.2 LY ≈ 25.2 trillion miles 3. Compare the radius and volume of the red giant Antares with those of the Sun. If Antares were placed at the center of our solar system, its radius would reach to between Mars and Jupiter. Volume: many estimates given; 60 million times Sun

Scoring Rubric 2 points: The student’s answer is an accurate and complete execution of the task or tasks. 1 point: The student’s answer contains attributes of an appropriate response but is flawed. 0 points: The student’s answer contains no attributes of an appropriate response.

Volume of Spheres 1170

LESSON

21.5

Name

Scale Factor

Class

Date

21.5 Scale Factor Essential Question: How does multiplying one or more of the dimensions of a figure affect its attributes?

Common Core Math Standards The student is expected to: COMMON CORE

Resource Locker

G-GMD.A.3

Explore

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Also G-GPE.B.7

Changes made to the dimensions of a figure can affect the perimeter and the area.

Mathematical Practices COMMON CORE

Exploring Effects of Changing Dimensions on Perimeter and Area 3

MP.2 Reasoning

2

To describe how changes in linear dimensions affect the area and

1

Language Objective -3

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Rex Features/AP Images

A

D (1, -1)

―

P = 18 + 4 √10

A=6

A = 18

Apply the transformation (x, y) → (x, 3y). Find the perimeter and the area.

Dimensions after (x, y) → (x, 3y)

Original Dimensions

―

_ P = 6 + 4√2

P = 6 + 4 √10

A=6

A = 18

Apply the transformation (x, y) → (3x, 3y). Find the perimeter and the area.

Dimensions after (x, y) → (3x, 3y)

_ P = 6 + 4√2

―

18 + 12 √2

P=

A=6 Module 21

A = 54 be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction

Lesson 5

1171

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Date Class

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Name

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Quest Essential

HARDCOVER PAGES 11711180

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COMMON CORE

Explore

IN2_MNLESE389847_U9M21L5.indd 1171

made Changes

figure sions of a to the dimen

y

3

2

Watch for the hardcover student edition page numbers for this lesson.

x 2

1

0

-2

C (3, 1)

B (0, 1)

1

-3

3

D (1, -1)

A (-2, -1)

-2 -3

one or more changing gate how and area. to investi perimeter Use the figure the figure affect its and the area. of perimeter dimensions . Find the

Apply the

transformat

→ (3x, y) ion (x, y)

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+4 P = 18

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Origi

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(x, nsions after

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P=6+4

A=

A=6

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. Find → (x, 3y) ion (x, y) transformat s Dimension Original _ √2

Apply the

transformat

→ (3x, ion (x, y)

Original

s

Dimension _ √2

A=6

Module 21

1L5.indd

47_U9M2

ESE3898

IN2_MNL

1171

―

y)

√10

18

―

A = 18

A=6

y) → (3x,

area. eter and the 3y) the perim y) → (x, s after (x, Dimension 6 + 4 √10 P=

P=6+4

Harcour t n Mifflin © Houghto Images /AP Features

Publishin

3y). Find

area. eter and the the perim 3y) y) → (3x, s after (x, Dimension √2 18 + 12 P=

―

P=6+4

Lesson 21.5

3

Dimensions after (x, y) → (3x, y)

_ P = 6 + 4√2

C

2

Apply the transformation (x, y) → (3x, y). Find the perimeter and the area.

PREVIEW: LESSON PERFORMANCE TASK

1171

C (3, 1)

-3

Original Dimensions

B

1

-2

Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area.

Original Dimensions

View the Engage section online. Discuss the photograph, and ask students to speculate on the purpose of the image. Then preview the Lesson Performance Task.

0

-2

A (-2, -1)

ENGAGE

When you multiply one dimension of a figure by a scale factor, the area is also multiplied by that factor. You cannot predict how the perimeter changes. When you multiply both dimensions of a figure by the same scale factor, the perimeter is multiplied by that factor and the area is multiplied by the square of that factor.

B (0, 1)

x

Explain to a partner the effect of a proportional dimension change on the area and perimeter of a geometric figure.

Essential Question: How does multiplying one or more of the dimensions of a figure affect its attributes?

y

A=

54

Lesson 5

1171 4/19/14

10:55 AM

4/19/14 10:55 AM

Reflect

EXPLORE

Describe the changes that occurred in Steps A and B. Did the perimeter or area change by a constant factor? Possible answer: When only one dimension was changed by a factor of 3, the area was

1.

Exploring the Effects of Changing Dimensions on Perimeter and Area

changed by a factor of 3. There was no consistent rule for the change in perimeter. Describe the changes that occurred in Step C. Did the perimeter or area change by a constant factor? Possible answer: When both dimensions were changed by a factor of 3, the perimeter

2.

INTEGRATE TECHNOLOGY

changed by a factor of 3 and the area changed by a factor of 3 2 or 9.

Explain 1

Students have the option of doing the Explore activity either in the book or online.

Describe a Non-Proportional Dimension Change

In a non-proportional dimension change, you do not use the same factor to change each dimension of a figure. Example 1

Find the area of the figure.

A = bh = 6 ⋅ 5 = 30 ft2

A = bh = 12 ⋅ 5 = 60 ft2

Find the area of the trapezoid. Then multiply the height by 0.5 and determine the new area. Describe the changes that took place. Original Figure

1 b +b h= A=_ ( 2) 2 1

_1(3 + 12)8 = 60 2

1 b +b h= Transformed Figure A = _ ( 2) 2 1

6 ft 3 in.

INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 If only one dimension of a geometric figure is

8 in.

_1(3 + 12)4 = 30 2

When the length of the trapezoid changes by a factor of

0.5 , the

12 in.

0.5 .

Reflect

3.

Discussion When a non-proportional change is applied to the dimensions of a figure, does the perimeter change in a predictable way? Possible answer: No. Since the change to the dimensions is non-proportional, not all of the lengths are altered. Therefore, the perimeter will not change by the factor of the change.

Your Turn

4.

changed by a factor of a, the effect on the perimeter is not as predictable as it is for area, which also changes by the same factor, a. This is because the perimeter total depends on which dimension is changed. For example, if a rectangle has dimensions of 2 in. and 3 in., doubling the 2 in. side makes the perimeter increase by 4 in. Doubling the 3 in. side, however, makes the perimeter increase by 6 in.

© Houghton Mifflin Harcourt Publishing Company

area of the trapezoid changes by a factor of

If all dimensions of a figure are changed by a factor of a, how does this change the area? the perimeter? The area changes by a factor of a 2; the perimeter changes by a factor of a.

5 ft

When the length of the parallelogram changes by a factor of 2, the area changes by a factor of 2.

QUESTIONING STRATEGIES

Find the area of the parallelogram. Then multiply the length by 2 and determine the new area. Describe the changes that took place. Original Figure Transformed Figure

Find the area of a triangle with vertices (-5, -2), (-5, 7), and (3, 1). Then apply the transformation (x, y) → (x, 4y) and determine the new area. Describe the changes that took place. 1 1( Possible answer: The original area is A = _ bh = __ 9 ⋅ 8) = 36. After the transformation the 2 2

EXPLAIN 1

1 1( area is A = _ bh = __ 36 ⋅ 8) = 144. When the base length changes by a factor of 4, the area 2 2

changes by a factor of 4.

Module 21

1172

Describe a Non-Proportional Dimension Change

Lesson 5

PROFESSIONAL DEVELOPMENT IN2_MNLESE389847_U9M21L5.indd 1172

Math Background The use of scale factors and scale drawings is an important application of mathematics. If a figure in the plane is transformed proportionally with a nonzero scale factor of k, then the area of the transformed figure is k 2 times the area of the original figure, and the perimeter is k times the perimeter of the original figure. If only one dimension is changed by a factor of k, then the area is changed by a factor of k, and the change in the perimeter is not predictable.

4/19/14 10:55 AM

QUESTIONING STRATEGIES If one dimension of a non-circular figure is changed by a factor of a, how does this change the area? The area changes by a factor of a.

Scale Factor

1172

Find the area of the figure. Then multiply the width by 5 and determine the new area. Describe the changes that took place.

5.

AVOID COMMON ERRORS

1 1( Possible answer: The original area is A = _ d d =_ 10 ⋅ 4) = 20. 2 1 2 2

Some students may think that changing one dimension by a factor of a also changes the perimeter by a factor of a. Point out that the change in perimeter is not predictable when only one dimension is changed. Have students verify this with examples.

1 1( After the transformation the area is A = _ d d =_ 50 ⋅ 4) = 100. 2 1 2 2

When the width changes by a factor of 5, the area changes by a factor of 5.

Explain 2

INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 As you present each concept, ask students

Find the circumference and area of the circle. Then multiply the radius by 3 and find the new circumference and area. Describe the changes that took place.

1173

Lesson 21.5

4

A = π(4) = 16π Transformed Figure C = 2π(12) = 24π 2 A = π(12) = 144π

The circumference changes by a factor of 3, and the area changes 2 by a factor of 9 or 3 .

© Houghton Mifflin Harcourt Publishing Company

Find the perimeter and area of the figure. Then multiply the length and height by __13 and find the new perimeter and area. Describe the changes that took place. Original Figure

Transformed Figure

P=

P=

4(6 √― 2 ) = 24 √― 2

1( 1 d d =_ 6 ⋅ 6) = 18 A= _ 2 2 1 2

The perimeter changes by a factor of

_1 3

A=

―

―

6

4(2 √2 ) = 8 √2

6

_1 d 1d 2 = _1(2 ⋅ 2) = 2 2

2

_1

__1

or 2 3 . , and the area changes by a factor of 9

Reflect

6.

Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally.

Possible answer:

side length of a square is changed, the size of the entire figure changes proportionally.

If the dimensions of a figure are changed proportionally, how does the area change? The area increases by the square of the dimension change.

C = 2π(4) = 8π

2

EXPLAIN 2

QUESTIONING STRATEGIES

Find the area and perimeter of a circle.

Original Figure

why they think each change in a dimension of a figure has a certain effect on its area. For instance, lead students to see that doubling the height of a rectangle doubles the rectangle’s area because b(2h) = 2(bh) = 2A.

INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Point out that if the radius of a circle or the

Describe a Proportional Dimension Change

In a proportional dimension change, you use the same factor to change each dimension of a figure. Example 2

Describe a Proportional Dimension Change

4 cm

10 cm

Effects of Changing Dimensions Proportionally Change in Dimensions

Perimeter or Circumference

Area

All dimensions multiplied by a

Changes by a factor of a

Changes by a factor of a 2

Module 21

1173

Lesson 5

COLLABORATIVE LEARNING IN2_MNLESE389847_U9M21L5.indd 1173

Peer-to-Peer Activity Have students work with partners to explore the effects of dimension changes. Have each student write two problems—one that shows a proportional dimension change to a triangle and another that shows a non-proportional dimension change to a parallelogram. Then have the partner find the area and perimeter of the new figure (if possible) and critique the solutions.

4/19/14 10:55 AM

Your Turn

7.

EXPLAIN 3

Find the circumference and area of the circle. Then multiply the radius by 0.25 and find the new circumference and area. Describe the changes that took place.

12

Describe a Proportional Dimension Change for a Solid

Possible answer: The original circumference is C = 2π(12) = 24π, 2 and the original area is A = π(12) = 144π. After the transformation, 2 ) ( the circumference is C = 2π 3 = 6π, and the area is A = π(3) = 9π. The circumference changes by a factor of 0.25, and the area changes 2 by a factor of (0.25) = 00625.

INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Point out that if a solid figure has a non-

Describe a Proportional Dimension Change for a Solid

Explain 3

In a proportional dimension change to a solid, you use the same factor to change each dimension of a figure.

proportional dimension change, then the volume changes by the product of the factors of each dimension change. The surface area does not have a predictable change, just as perimeter does not have a predictable change for a two-dimensional figure.

Find the volume of the composite solid.

Example 2

A company is planning to create a similar version of this storage tank, a cylinder with hemispherical caps at each end. Find the volume and surface area of the original tank. Then multiply all the dimensions by 2 and find the new volume and surface area. Describe the changes that took place. The volume of the solid is V = πr 2h + __34 πr 3, and the surface area is A = 2πrh + 4πr 2.

Transformed Solid 2 4 π(6) 3 = 1152π cu. ft. V = π(6) (24) + _ 3 2 SA = 2π(6 ⋅ 24) + 4π(6) = 432π sq. ft.

2 4 π(3) 3 = 144π cu. ft. V = π(3) (12) + _ 3 2 SA = 2π(3 ⋅ 12) + 4π(3) = 108π sq. ft.

The volume changes by a factor of 8, and the surface area changes by a factor of 4.

A children’s toy is shaped like a hemisphere with a conical top. A company decides to create a smaller version of the toy. Find the volume and surface area of the original toy. Then multiply all dimensions by __23 and find the new volume and surface area. Describe the changes that took place.

4 in.

The volume of the solid is V = __13 πr 2h + __23 πr 3, _

and the surface area is A = πr√r 2 + h 2 + 2πr 2.

3 in.

Original Solid

_1 π(3)24 + _2 π(3)3 = 18π

V= 3

3

A = π(3) √――― 3 + 4 + 2π(3) 2

2

2

8 __ 27

()

2 ( )3 __ 1 ( )2 _ π 2 83 + _ π 2 = 16 π cu. in. V= _ 3 3 3

= 33π sq. in.

The volume changes by a factor of Module 21

Transformed Solid cu. in.

A = π(2)

――――

If the dimensions of a solid figure are changed proportionally, how does the volume change? How does the surface area change? The volume increases by the cube of the dimension change; the surface area increases by the square of the dimension change.

√(2) + (_83) + 2π(2) = __443 π sq. in. 2

2

, and the surface area changes by a factor of 1174

QUESTIONING STRATEGIES

12 ft © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Ocean/ Corbis

Original Solid

6 ft

2

_4 9

. Lesson 5

DIFFERENTIATE INSTRUCTION IN2_MNLESE389847_U9M21L5.indd 1174

Modeling

4/19/14 10:55 AM

Have students use blocks to study how the volume of three-dimensional figures changes if there is a change to the dimensions. Have student groups create a rectangular prism with blocks and give the volume, which equals the number of blocks. Then have them increase the dimensions of the prism proportionally and count the blocks (the volume increases by the third power of the dimension change). Ask them also to increase only one dimension of the original prism and explore how the volume changes.

Scale Factor

1174

Reflect

ELABORATE

8.

QUESTIONING STRATEGIES

Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally.

Possible answer:

Effects of Changing Dimensions Proportionally

How does changing the dimensions of a figure proportionally by a factor of k affect the area? The area is changed by a factor of k 2. What does a change to the area of a two-dimensional figure by a factor of k mean in terms of the perimeter? You cannot determine how it changes the perimeter.

Change in Dimensions

Surface Area

Volume

All dimensions multiplied by a

Changes by a factor of a 2

Changes by a factor of a 3

Your Turn

9.

A farmer has made a scale model of a new grain silo. Find the volume and surface area of the model. Use the scale ratio 1 : 36 to find the volume and surface area of the silo. Compare the volumes and surface areas relative to the scale ratio. Be consistent with units of measurement.

3 in.

1 2 The volume of the solid is V = _ πr h + πr 2h (same height for cylinder 3

―――

SUMMARIZE THE LESSON

and cone), and the surface area is A = πr √r 2 + h 2 + 2πrh + πr 2. For

What happens to a figure’s area when a dimension of the figure is changed? What happens when two dimensions are changed? one dimension: area changes by the same factor as the dimension change and perimeter is not predictable; two dimensions: area changes by the square of the dimension change; perimeter changes by the same factor as the dimension change

the model, the volume is 64π cu. in., and the surface area is 60π sq. in.

3 in. 8 in.

For the silo, the volume is 2985984π cu. in. and the surface area is 77760π sq. in. The volume changes by a factor of 46656 = 36 3, and the surface area changes by a factor of 1296 = 36 2.

© Houghton Mifflin Harcourt Publishing Company

Elaborate 10. Two square pyramids are similar. If the ratio of a pair of corresponding edges is a : b, what is the ratio of their volumes? What is the ratio of their surface areas? The ratio of their volumes is a 3 : b 3, and the ratio of their surface areas is a 2 : b 2.

11. Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change? Possible answer: With non-proportional dimension changes, the effect on perimeter, area,

and volume may not be clearly defined by a scale factor.

Module 21

1175

Lesson 5

LANGAUGE SUPPORT IN2_MNLESE389847_U9M21L5.indd 1175

Connect Vocabulary To help students understand what a proportional dimension change is, define proportional as having the same ratio, and a dimension as a measurement of length in one direction. So, a proportional dimension change is a change in each length measurement such that the lengths maintain the same ratio to each other.

1175

Lesson 21.5

4/19/14 10:55 AM

Evaluate: Homework and Practice

EVALUATE

A trapezoid has the vertices (0, 0), (4, 0), (4, 4), and (-3, 4). 1.

Describe the effect on the area if only the x-coordinates of the vertices are multiplied by __12 .

2.

Describe the effect on the area if only the y-coordinates of the vertices are multiplied by __12 .

Original Figure

Original Figure

(0, 0), (4, 0), (4, 4), and (-3, 4) (b 1 + b 2)h _ (4 + 7)4 _

(0, 0), (4, 0), (4, 4), and (-3, 4)

= 2 Transformed Figure A=

A= 12

=

2

11 1 __ =_

_ _

(0, 0), (4, 0), (4, 2), and (-3, 2) (b 1 + b 2)h _ (4 + 7)2 _

= 11

2

A=

2

1 The area is multiplied by _ .

11 1 __ =_

2

Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by __12 .

22

4.

(b 1 + b 2)h (______ 4 + 7 )4 = = 22 A = _______ 2 2

44

22

=

2

(0, 0), (8, 0), (8, 2), and (-6, 2) (8 + 14)2 b 1 + b 2)h (_______ _

= 5.5

A=

22 __ =1

4

2

=

2

= 22

22

1 The area is multiplied by _ . 4

The area doesn’t change. It would have been doubled by the change in the x-coordinates and halved by the change in the y-coordinates.

Module 21

Exercise

Depth of Knowledge (D.O.K.)

COMMON CORE

Mathematical Practices

1 Recall of Information

MP.5 Using Tools

2–4

3 Strategic Thinking

MP.2 Reasoning

5–12

3 Strategic Thinking

MP.5 Using Tools

1 Recall of Information

MP.2 Reasoning

14–16

2 Skills/Concepts

MP.4 Modeling

17–18

1 Recall of Information

MP.5 Using Tools

19–21

3 Strategic Thinking

MP.2 Reasoning

1

13

Explore Exploring Effects of Changing Dimensions on Perimeter and Area

Exercises 1–4, 15

Example 1 Describe a Non-Proportional Dimension Change

Exercises 5–8, 13, 20–21

Example 2 Describe a Proportional Dimension Change

Exercises 9–10 , 13–14, 16–18, 22–23

Example 3 Describe a Proportional Dimension Change for a Solid

Exercises 11–12, 19, 24

below to confirm that there is a pattern in the effects on a figure when one or more dimensions is changed. Factor of k

Lesson 5

1176

IN2_MNLESE389847_U9M21L5.indd 1176

Practice

INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Have students fill out a table like the one

© Houghton Mifflin Harcourt Publishing Company

Transformed Figure

(0, 0), (2, 0), (2, 2), and (-1.5, 2) 2

Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by __12 .

Concepts and Skills

Original Area

_ _

A=

2

(0, 0), (4, 0), (4, 4), and (-3, 4)

Original Area (b 1 + b 2)h (4 + 7)4 = = 22 A= 2 2 Transformed Figure

5.5 11 1 ___ = __ =_

= 11

2

Original Figure

(0, 0), (4, 0), (4, 4), and (-3, 4)

2 + 3.5)2 (_______

=

2

1 The area is multiplied by _ . 2

Original Figure

b 1 + b 2)h (_______

ASSIGNMENT GUIDE

Original Area (b 1 + b 2)h (4 + 7)4 = = 22 A= 2 2 Transformed Figure

= 22

2

(0, 0), (2, 0), (2, 4), and (-1.5, 4) (b 1 + b 2)h _ (2 + 3.5)4 _

3.

• Online Homework • Hints and Help • Extra Practice

Proportional Dimension Change

Non-Proportional Dimension Change

Area

Factor of k2

Factor of k

Perimeter

Factor of k

Not predictable

Volume

Factor of k

3

Not predictable

Surface Area

Factor of k2

Not predictable

4/19/14 10:55 AM

Scale Factor

1176

Describe the effect of the change on the area of the given figure.

INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Point out that there may be more than one

5.

The height of the triangle is doubled.

Original Figure bh ⋅ 21 _____ A = __ = 12 = 126 m 2 2 2

6.

12 m

Original Figure

(b 1 + b 2)h (_______ 12 + 8)5 A = _______ = = 50 cm 2 2 2

21 m

Transformed Figure 24 ⋅ 21 = 252 m 2 bh A = __ =_ 2 2 252 ___ = 2 126

way to produce a certain change in area. For example, suppose you want to double the area of a rectangle. You can do this by doubling the length, doubling the width, or by increasing the length and width proportionally by a factor of √2 .

Transformed Figure 5 12 + 8)_ (b 1 + b 2)h (_______ 100 50 3 A = _______ = = ___ = __ cm 2 6 2 2 3 50 1 _ ÷ 50 = _ 3 3 1 The area is multiplied by _ . 3

The area is doubled.

―

7.

The base of the parallelogram is multiplied by __23 .

Original Figure A = bh = 24 ⋅ 9 = 216 in

8.

9 in. 24 in.

2

Transformed Figure

Communicate Mathematical Ideas A triangle has vertices (1, 5), (2, 3), and (-1, -6) . Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle, without doing any calculations. Explain. The area is mulitplied by 4. When only one dimension of a figure is multiplied by a factor, the area of the figure is multiplied by the same factor.

A = bh = 16 ⋅ 9 = 144 in 2

144 2 ___ =_ 216

The height of a trapezoid with base lengths 12 cm and 18 cm and height 5 cm is multiplied by __13 .

3

2 The area is multiplied by _ . 3

Describe the effect of each change on the perimeter or circumference and the area of the given figure. 9.

The base and height of an isosceles triangle with base 12 in. and height 6 in. are both tripled.

Original Figure bh ⋅6 ____ = 12 = 36 in 2 A = __ 2 2

© Houghton Mifflin Harcourt Publishing Company

P = 8.5 + 8.5 + 12 = 29 in Transformed Figure

36 ⋅ 18 bh A = __ = _ = 324 in 2 2 2 324 87 _ Change in Perimeters __ = 3 Change in Areas =9 29 36 The perimeter is tripled. The area is multiplied by 9.

P = 25.5 + 25.5 + 36 = 87 in

18 ft

10. The base and height of the rectangle are both multiplied by __12 .

6 ft

Original Figure P = 2(b + h) = 2(18 + 6) = 48 ft

A = bh = 18 ⋅ 6 = 108 ft 2

Transformed Figure P = 2(b + h) = 2(9 + 3) = 24 ft 24 1 Change in Perimeters __ =_ 48 2

A = bh = 9 ⋅ 3 = 27 ft 2

27 1 Change in Areas ___ =_ 4 108 1 1 _ The perimeter is multiplied by . The area is multiplied by __. 4

2

Module 21

Exercise

IN2_MNLESE389847_U9M21L5.indd 1177

1177

Lesson 21.5

Lesson 5

1177

Depth of Knowledge (D.O.K.)

COMMON CORE

Mathematical Practices

22

2 Skills/Concepts

MP.4 Modeling

23

2 Skills/Concepts

MP.4 Modeling

24

3 Strategic Thinking

MP.6 Precision

4/19/14 10:55 AM

3. 12. The dimensions are multiplied by _ 5

11. The dimensions are multiplied by 5.

AVOID COMMON ERRORS Students may make errors when finding the effects of proportional dimension changes on circles. Point out that because there is only one variable in the formula for a circle, changing the dimension is equivalent to doing a proportional dimension change. The area of the new circle increases by the square of the dimension change to the radius.

2 yd 10 m

3 yd 5m

Original: V = πr h = 12π yd ;

Original: V = ℓwh = 250 m 3;

Transformed: V = πr 2h = 1500π yd 3; 1500π = 125; change in volumes 12π volume is multiplied by 125.

Transformed: V = ℓwh = 54 m 3; 54 27 = change in volumes ; 250 125 27 . volume is multiplied by 125

2

3

_ _ _

_

13. For each change, check whether the change is non-proportional or proportional. A. The height of a triangle is doubled.

proportional

non-proportional

B. All sides of a square are quadrupled.

proportional

non-proportional

proportional

non-proportional

proportional

non-proportional

proportional

non-proportional

3. C. The length of a rectangle is multiplied by _ 4 D. The height of a triangular prism is tripled.

―

E. The radius of a sphere is multiplied by √5 .

14. Tina and Kleu built rectangular play areas for their dogs. The play area for Tina’s dog is 1.5 times as long and 1.5 times as wide as the play area for Kleu’s dog. If the play area for Kleu’s dog is 60 square feet, how big is the play area for Tina’s dog?

Possible dimensions of play area: 10 feet by 6 feet; New Area: A = ℓw = 135 ft 2

15. A map has the scale 1 inch = 10 miles. On the map, the area of Big Bend National Park in Texas is about 12.5 square inches. Estimate the actual area of the park in acres. (Hint: 1 square mile = 640 acres)

Possible map dimensions 2.5 in. by 5 in.; New Area: A = ℓw = 1250 mi 2; ≈ 800,000 acres

_

17. Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled. Find the perimeter of the new triangle in inches. Possible triangle dim. 6 in. by 6 in. by 6 in.; New Perimeter: P = 3(12) = 36 in.

A rectangular prism has vertices (0, 0, 0), (0, 3, 0), (7, 0, 0), (7, 3, 0), (0, 0, 6), (0, 3, 6), (7, 0, 6) and (7, 3, 6).

© Houghton Mifflin Harcourt Publishing Company

16. A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $36.75 per week. The cost of each ad is based on its area. If the owner of the restaurant decided to double the width and height of the ad, how much will the new ad cost? 32 Original: A = bh = 8 in 2; New: A = bh = 32 in 2; = 4; since the area is 4 times as large, it 8 will cost $36.75 × 4 = $147.00.

18. Suppose all the dimensions are tripled. Find the new vertices.

(0, 0, 0), (0, 9, 0), (21, 0, 0), (21, 9, 0), (0, 0, 18), (0, 9, 18), (21, 0, 18), and (21, 9, 18) 19. Find the effect of the change on the volume of the prism. 3402 = 27; the volume is multiplied by 27. Original: V = ℓwh = 126; New: V = ℓwh = 3402; 126

_

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Scale Factor

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20. How would the effect of the change be different if only the height had been tripled? 378 = 3; the volume would have been Original: V = ℓwh = 126; New: V = ℓwh = 378; 126 multiplied by 3 instead of 27.

COLLABORATIVE LEARNING

_

Divide students into groups. Have each group member find the effect of doubling all dimensions to the area and perimeter of a triangle, circle, parallelogram, and kite. Have them verify their results using algebra and share their results within the group.

21. Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram? if the parallelogram does have to be similar to the original parallelogram?

―

Multiply the base or height by 5; Multiply the base and height by √5 .

H.O.T. Focus on Higher Order Thinking

JOURNAL

22. Algebra A square has a side length of (2x + 5) cm.

Have students describe the effects of a proportional dimension change on the area and perimeter of a two-dimensional figure, and on the volume and surface area of a three-dimensional figure. Ask them to include examples.

a. If the side length is mulitplied by 5, what is the area of the new square?

5(2x + 5) = 10x + 25; A = s 2 = (10x + 25) = (100x 2 + 500x + 625)cm 2 2

b. Use your answer to part (a) to find the area of the original square without using the area formula. Justify your answer.

100x + 500x + 625 __ = 4x 2

2 + 20x + 25; (4x 2 + 20x + 25) cm 2; multiplying the side 25 length of a square by 5 multiplies the area of the square by 5 2 = 25, so dividing the

area found in part (a) by 25 gives the original area. 23. Algebra A circle has a diameter of 6 in. If the circumference is multiplied by (x + 3), what is the area of the new circle? Justify your answer.

If the circumference is multiplied by (x + 3), then so is the radius. The original radius is 3 in.

A = πr 2 = π(3x + 9) ; (9πx 2 + 54πx + 81π) in 2; If the circumference is multiplied by (x + 3),

© Houghton Mifflin Harcourt Publishing Company

2

then so is the radius. The original radius is 3 in., so the new radius is 3(x + 3) = (3x + 9) in. 24. Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism. Make a conjecture about the ratio of the surface area of the new prism to its volume. Test your conjecture using a cube with an edge length of 1 and a scale factor of 2.

1 For a scale factor of k, the ratio of surface area to volume of the new prism is __ times k

the ratio of surface area to volume of the old prism. Original: SA = 1 2 ∙ 6 = 6; V = 1 3 = 1; 6 3 6 SA SA 24 3 = ; SA = 2 2 ∙ 6 = 24; V = 2 3 = 8; = = ; ∙ 1 = New: 1 1 1 2 8 1 V V

_ _

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_ _ __ _ _

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Lesson Performance Task

INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Question 3 in the Lesson Performance Task

On a computer screen, lengths and widths are measured not in inches or millimeters but in pixels. A pixel is the smallest visual element that a computer is capable of processing. A common size for a large computer screen is 1024 × 768 pixels. (Widths rather than heights are conventionally listed first.) For the following, assume you’re working on a 1024 × 768 screen.

1. You have a photo measuring 640 × 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen. Find the measurements of the photo after it has been scaled up. Explain how you found the answer.

establishes that twelve 256 × 256 pixel photos can be placed on a 1024 × 768 pixel screen with no overlap and no gaps between photos. Find the sizes of other square photos larger than 50 × 50 pixels that can be placed on a 1024 × 768 pixel screen with no overlap and no gaps between photos. For each measurement that you find, give the number of rows, the number of squares in each row, and the total number of photos. Sample answers: 128 × 128 pixels: 6 rows of 8 photos each, 48 photos total; 64 × 64 pixels: 12 rows of 16 photos each, 192 photos total

1024 pixels

768 pixels

2. a. Explain why you can’t enlarge the photo proportionally so that it is as tall as the computer screen. b. Why can’t you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 ÷ 640 and the height by a factor of 768 ÷ 300? 3. You have some square photos and you would like to fill the screen with them, so there is no overlap and there are no gaps between photos. Find the dimensions of the largest such photos you can use (all of them the same size), and find the number of photos. Explain your reasoning.

1. 1024 × 480; The photo must be scaled up in width by a factor of 1024 ÷ 640 = 1.6. So, apply the same factor to the height: 300 × 1.6 = 480.

INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 A coordinate grid appears on a computer

2. a. You would have to scale the height of the photo up by a factor of 768 ÷ 300 = 2.56. Scaling the width by the same factor would create a photo that is 640 × 2.56 = 1638.4 pixels in width, which exceeds the width of the screen. b. You would increase the width by a factor of 1.6 and the height by a factor of 2.56, creating a distorted image.

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screen. A square on the grid has vertices at (-4, 4), (4, 4), (4, -4), and (-4, -4). A Web designer leaves the grid unchanged but scales up the square by a factor of 1.5 vertically and 0.8 horizontally.

© Houghton Mifflin Harcourt Publishing Company

3. 256 × 256. Sample answer: The height of the photos must be a factor of 768 pixels so that the photos fill the screen vertically. Two vertical squares would each have heights of 768 ÷ 2 = 384 pixels. But since 384 is not a factor of 1024, photos measuring 384 × 384 would not exactly fill the screen horizontally. Three vertical squares would each have heights of 768 ÷ 3 = 256 pixels. Since 256 is a factor of 1024, photos measuring 256 × 256 would fill the screen horizontally. The screen would have 3 rows of 4 photos each for a total of twelve 256 × 256 photos.

a. What are the vertices of the new rectangle? (-3.2, 6), (3.2, 6), (3.2, -6), and (-3.2, -6) b. By what factor has the area of the rectangle been changed? 1.2

Lesson 5

EXTENSION ACTIVITY IN2_MNLESE389847_U9M21L5.indd 1180

Computer screen resolutions are sometimes referred to by a set of acronyms, each with a corresponding size in pixels. Ask students to choose three of the following acronyms to research: VGA, SVGA, XGA, SXGA, and WUXGA. Have them find out the dimensions of each of their choices, then create a scale drawing on graph paper that relates the different sizes. Invite students to share their results with the class. VGA: 640 × 480; SVGA: 800 × 600; XGA: 1024 × 768; SXGA: 1280 × 1040; WUXGA: 1920 × 1200

4/19/14 10:55 AM

Scoring Rubric 2 points: The student’s answer is an accurate and complete execution of the task or tasks. 1 point: The student’s answer contains attributes of an appropriate response but is flawed. 0 points: The student’s answer contains no attributes of an appropriate response.

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MODULE

21

MODULE

STUDY GUIDE REVIEW

21

Volume Formulas

Study Guide Review

Essential Question: How can you use volume formulas to solve real-world problems?

ASSESSMENT AND INTERVENTION

KEY EXAMPLE

Key Vocabulary

right prism (prisma recto) right cylinder (cilindro recto)

(Lesson 21.1)

Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters. Write an exact answer. V = πr 2h

Write the formula for the volume of a cylinder.

= π(3) (5) 2

= 45π cm 3

Assign or customize module reviews.

Substitute. Simplify.

KEY EXAMPLE

1 Bh V=_ 3 1 (12) 2(7) =_ 3 = 336 m 3

COMMON CORE

Mathematical Practices: MP.1, MP.2, MP.3, MP.4, MP.6 G-GMD.A.3, G-MG.A.1

• How to find the volume of the sinkhole: Students can use the volume formula for a right cylinder, V = πr 2h. The volume is V = (π33 2)(100) ≈ 342, 119 cubic feet.

Substitute. Simplify. (Lesson 21.3)

Find the volume of a cone with a base diameter of 16 feet and a height of 18 feet. Write an exact answer. © Houghton Mifflin Harcourt Publishing Company

• The dimensions of the sinkhole: Students can research this and use average numbers, or you can give them an estimate of 66 feet wide by 100 feet deep.

Write the formula for the volume of a pyramid.

KEY EXAMPLE

SUPPORTING STUDENT REASONING

• The shape of the sinkhole: Students can assume that the sinkhole is a cylinder. This shape should give a reasonable estimate of the volume of the hole.

(Lesson 21.2)

Find the volume of a square pyramid with a base side length of 12 inches and a height of 7 inches.

MODULE PERFORMANCE TASK

Students should begin this problem by focusing on what information they will need. Here are some issues they might bring up.

oblique prism (prisma oblicuo) oblique cylinder (cilindro oblicuo) cross section (sección transversal)

1 (16 ft) r=_ 2 = 8 ft 1 πr 2h V=_ 3 1 π(8) 2(18) =_ 3 = 384π ft 3

Find the radius. Simplify. Write the formula for the volume of a cone. Substitute. Simplify.

KEY EXAMPLE

(Lesson 21.4)

Find the volume of a sphere with a radius of 30 miles. Write an exact answer. 4 πr 3 V=_ 3 4 π(30) 3 =_ 3 = 36,000 π mi 3

Write the formula for the volume of a sphere. Substitute. Simplify.

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SCAFFOLDING SUPPORT

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• The cost of a cubic foot of concrete: Students can research this or use $4 per cubic foot.

• The formula for the volume of a cylinder is V = πr h, where r is the radius and h is the height.

• Other costs for the repair: Costs will include concrete, labor, and material transportation.

• Encourage students to use unit analysis when they calculate the material cost to fill the sinkhole. The units of cubic feet should cancel, and $ should remain in the numerator.

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2

EXERCISES Find the volume of each figure. Write an exact answer. (Lessons 21.1–21.4) 225

1.

2.

Assumptions:

6.3

9 10

SAMPLE SOLUTION 621.81

The sinkhole is in the shape of a right cylinder, with a diameter of 66 feet and a height of 100 feet.

4.7

5

21

8448π cm 3

3.

16 cm

The cost of concrete is $4 per cubic foot.

24 ft 3

3.6 ft

4.

Method:

33 cm

Find the volume of the sinkhole using V = πr 2h, which is the volume of material needed to fill it. Then multiply the result by $4/ft 3 to find the total cost.

4 ft 5 ft

24π m 3

5. 8m

288π

6.

Find the volume.

12

V = πr 2h = π(33 2)(100) ≈ 342, 119 ft 3 3m

7.

― ― area: 648 √3 in . The perimeter is multiplied by 3. The area is multiplied by 9.

Original perimeter: 48 in.; original area: 72 √3 in 2 ; new perimeter: 144 in.; new 2

MODULE PERFORMANCE TASK

How Big Is That Sinkhole? In 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it. The sinkhole has an estimated depth of about 100 feet. How much material is needed to fill the sinkhole? Determine what information is needed to answer the question. Do you think your estimate is more likely to be too high or too low? What are some material options for filling the sinkhole, and how much would they cost? Which material do you think would be the best choice?

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© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Stringer/ Reuters/Corbis

One side of a rhombus measures 12 inches. Two angles measure 60°. Find the perimeter and area of the rhombus. Then multiply the side lengths by 3. Find the new perimeter and area. Describe the changes that took place. (Lesson 21.5)

The estimate is likely to differ from the actual amount, but it is difficult to determine whether it is an overestimate or underestimate. The shape of the sinkhole will not be a perfect right cylinder, and the depth measurements will vary depending on where the measurement is taken. Find the total cost.

$4 ≈ $1, 368,500 (342, 199 ft 3) ____ 3

1 ft If concrete is used, it will cost well over a million dollars in materials alone to fill the sinkhole.

Study Guide Review

DISCUSSION OPPORTUNITIES

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• What other considerations may need to be made when planning to repair the sinkhole? • How should the loss of a building also impact how this real-world problem is solved? Assessment Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain. 0 points: Student does not demonstrate understanding of the problem.

Study Guide Review 1182

Ready to Go On?

Ready to Go On?

21.1–21.5 Volume Formulas

ASSESS MASTERY Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.

• Online Homework • Hints and Help • Extra Practice

Find the volume of the figure. (Lessons 21.1–21.4) An oblique cylinder next to a cube.

1.

10 ft

A prism of volume 3 with a pyramid of the same height cut out.

2.

44 ft

ASSESSMENT AND INTERVENTION

c

b a

Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.

= 2400π + 8000 A cone with a square pyramid of the same height cut out. The pyramid has height l, and its square base has area l 2.

3.

ADDITIONAL RESOURCES • Reteach Worksheets Differentiated Instruction Resources • Reading Strategies • Success for English Learners • Challenge Worksheets Assessment Resources • Leveled Module Quizzes

A cube with sides of length s with the biggest sphere that fits in it cut out.

4.

()

4 π __ s s 3 - __ 3 2

Top View © Houghton Mifflin Harcourt Publishing Company

Response to Intervention Resources

2 Bh = __ 2 ·3=2 1 Bh = __ Bh - __ 3 3 3

2 3 πr 2h + l 3 = π(10) (44 - 20) + (20)

3

(

)

π 4 π __ s3 = s 3 1 - __ = s 3 - __ 6 3 8

l√2

__1 πr h - __1 l h = __1 h(πr 2

3

3

2

3

(

2

⎡

( ―) - l ⎥

l √2 1 l π _____ - l 2) = __ 3 ⎢ 2 ⎣

) (

⎤

2

2

⎦

)

π -1 1 l π__ l 3 __ l 2 - l 2 = __ = __ 3 3 2 2 ESSENTIAL QUESTION

How would you find the volume of an ice-cream cone with ice cream in it? What measurements would you need?

5.

Answers may vary. Sample: An ice-cream cone is composed approximately of a semi-sphere and a cone. You need the radius of the cone and the height of the cone. Use the volume of a cone formula and add it to the volume of the sphere divided by 2: Module 21

COMMON CORE

_1 πr h + _1 (_4 πr ). You can then simplify. 2

3

3

2 3

Study Guide Review

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Common Core Standards

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Content Standards Mathematical Practices

Lesson

Items

21.1

1

G-GMD.A.3

MP.4

21.1, 21.2

2

G-GMD.A.3

MP.4

21.2, 21.3

3

G-GMD.A.3

MP.4

21.1, 21.4

4

G-GMD.A.3

MP.4

MODULE MODULE 21 MIXED REVIEW

MIXED REVIEW

Assessment Readiness 1. A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism, as shown. The volume of the monument is 66 cubic feet. Determine whether the given measurement could be the height of the monument. Select Yes or No for A–C. A. 10 feet Yes No B. 13 feet C. 15 feet

Yes Yes

Assessment Readiness ASSESSMENT AND INTERVENTION

10 ft

No No

2. A standard basketball has a radius of about 4.7 inches. Choose True or False for each statement. A. The diameter of the basketball is about 25 inches. True B. The volume of the basketball is approximately 277.6 in 3. True C. The volume of the basketball is approximately 434.9 in 3. True

3 ft

2 ft

Assign ready-made or customized practice tests to prepare students for high-stakes tests.

False False

ADDITIONAL RESOURCES

False

Assessment Resources • Leveled Module Quizzes: Modified, B

3. A triangle has a side of length 8, a second side of length 17, and a third side of length x. Find the range of possible values for x. 9 < x < 25

3m

Sample Answer: The volume of the figure is 11 cm 226.2 m 3, or 72π m 3. The right triangle composed of the height of the cone, the radius of the cylinder, and the slant height of the cone is a special right triangle, so the height of the cone is 3 meters. The height of the cylinder is the length of the composite figure minus the radius of the hemisphere and the height of the cone, so it is 11 - 3 - 3 = 5 meters.

COMMON CORE

AVOID COMMON ERRORS

3√2 m

Item 1 Some students will be confused that the question asks for the height when the height of the rectangular prism is given. Point out that the height of the monument includes both shapes in the composite figure, not just the prism.

© Houghton Mifflin Harcourt Publishing Company

4. Find the approximate volume of the figure at right, composed of a cone, a cylinder, and a hemisphere. Explain how you found the values needed to compute the volume.

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Study Guide Review

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Common Core Standards

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Content Standards Mathematical Practices

Lesson

Items

21.1, 21.2

1

G-GMD.A.3

MP.2

21.4

2

G-GMD.A.3

MP.1, MP.2

IM1 21.3

3*

G-MG.A.3

MP.6

21.1, 21.3, 18.3

4*

G-GMD.A.3

MP.2

* Item integrates mixed review concepts from previous modules or a previous course.

Study Guide Review 1184

UNIT

9

UNIT 9 MIXED REVIEW

Assessment Readiness

MIXED REVIEW

Assessment Readiness

1. Consider each congruence theorem below. Can you use the theorem to determine whether 4ABC ≅ 4ABD?

ASSESSMENT AND INTERVENTION

40°

C

40°

No No No

2. For each pyramid, determine whether the statement regarding its volume is true. Select True or False for each statement. A. A rectangular pyramid with ℓ = 3 m, w = 4 m, h = 7 m has volume 84 m 3. True False B. A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 ft 2. True False C. A pyramid with the same base and height of a prism has less volume. True False

Assessment Resources • Leveled Unit Tests: Modified, A, B, C • Performance Assessment

© Houghton Mifflin Harcourt Publishing Company

AVOID COMMON ERRORS

3. For each shape, determine whether the statement regarding its volume is true. Select True or False for each statement. A. A cone with base radius r = 5 in. and h = 12 in. has volume 100π in3. True False 6 B. A sphere with radius r = _ m has π 8 volume 2 m 3. True False π C. A sphere is composed of multiple cones with the same radius. True False

_

4. DeMarcus draws 4ABC. Then he translates it along the vector (-4, -3), rotates it 180°, and reflects it across the x-axis. y Choose True or False for each statement. 4 A. The final image of 4ABC is in Quadrant IV. True False B. The final image of 4ABC is a right triangle. True False C C. DeMarcus will get the same result if he True False 0 performs the reflection followed by the translation and rotation. Unit 9

COMMON CORE IN2_MNLESE389847_U9UC 1185

A B

x 4

1185

Common Core Standards

Items

Unit 9

Yes Yes Yes

B. SAS Triangle Congruence Theorem C. SSS Triangle Congruence Theorem

ADDITIONAL RESOURCES

1185

D

B

Select Yes or No for A–C. A. ASA Triangle Congruence Theorem

Assign ready-made or customized practice tests to prepare students for high-stakes tests.

Item 2 Some students will forget the last step of finding the volume of a pyramid or cone, failing to 1 . Encourage students to double-check multiply by _ 3 the formulas to ensure they have included every step.

A

• Online Homework • Hints and Help • Extra Practice

Content Standards Mathematical Practices

1*

G-SRT.C.8

MP.5

2

G-GMD.A.3

MP.2

3

G-GMD.A.3

MP.2

4*

G-GPE.B.4, G-CO.A.4

MP.6

5

G-GMD.A.3

MP.2

6

G-GMD.A.3

MP.1

* Item integrates mixed review concepts from previous modules or a previous course.

4/19/14 9:14 AM

5. A volleyball has a radius of about 8 inches. A soccer ball has a radius of about 4.25 inches. Determine whether each statement regarding the volume of each ball is true. Select True or False for each statement. A. The volume of the volleyball is about 682.7π in.3 True False B. The volume of the soccer ball is about 76.8π in.3

True

False

C. The volume of the volleyball is about 3.75π times more than the volume of the soccer ball.

True

False

PERFORMANCE TASKS There are three different levels of performance tasks: * Novice: These are short word problems that require students to apply the math they have learned in straightforward, real-world situations. ** Apprentice: These are more involved problems that guide students step-by-step through more complex tasks. These exercises include more complicated reasoning, writing, and open ended elements.

6. A cone and a cylinder have the same height and base diameter. Is each statement regarding the volume of each shape true? Select True or False for each statement. A. If the height is 8 cm and the base diameter is 6 cm, the volume of the cone is 72π cm3. True False B. If the height is 6 cm and the base diameter is 4 cm, the volume of the cylinder is 24π cm3. True False C. The volume of the cylinder is always 3 times the volume of the cone. True False

***Expert: These are open-ended, nonroutine problems that, instead of stepping the students through, ask them to choose their own methods for solving and justify their answers and reasoning.

7. A vase is in the shape of a cylinder with a height of 15 feet. The vase holds 375π in.3 of water. What is the diameter of the base of the vase? Show your work. V = πr2h

375π = πr2(15) 25 = r2 r=5

d = 2(5) = 10 in. 8. A salt shaker is a cylinder with half a sphere on top. The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram.

SALT

99π cm3 or about 311.02 cm3; I found the volume of a cylinder with a base radius of 3 cm and a height of 8 cm. Then, I found the volume of a sphere with a radius of 3 cm. I added the volume of the cylinder to half the volume of the sphere.

9 cm

3 cm

9. A cube is dilated by a factor of 4. By what factor does its volume increase? Explain your reasoning.

© Houghton Mifflin Harcourt Publishing Company

What is the volume of the salt shaker? Explain how you got your answer.

The volume increases by a factor of 64. When the cube is dilated, each side length is increased by the dilation factor, so the volume is increased by the cube of the dilation factor.

Unit 9

1186

COMMON CORE

Common Core Standards

IN2_MNLESE389847_U9UC 1186

Items

4/19/14 9:14 AM

Content Standards Mathematical Practices

7*

G-GMD.A.3

MP.1

8

G-GMD.A.3

MP.6

9

G-SRT.A.1

MP.3, MP.8

* Item integrates mixed review concepts from previous modules or a previous course.

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Performance Tasks

SCORING GUIDES

10. A scientist wants to compare the volumes of two cylinders. One is twice as high and has a diameter two times as long as the other. If the volume of the smaller cylinder is 30 cm 3, what is the volume of the larger cylinder?

Item 10 (2 points) Award the student 1 point for the correct scale factor, and 1 point for the correct volume.

We know that the volume of the smaller cylinder is πr2h = 30 cm3. The larger cylinder can be represented by π(2r)2(2h), which simplifies to 8(πr 2h). Substituting, the larger cylinder has a volume of 8(30) = 240 cm3.

Item 11 (6 points)

11. You are trying to pack in preparation for a trip and need to fit a collection of children’s toys in a box. Each individual toy is a composite figure of four cubes, and all of the toys are shown in the figure. Arrange the toys in an orderly fashion so that they will fit in the smallest box possible. Draw the arrangement. What is the volume of the box if each of the cubes have side lengths of 10 cm?

3 points for drawing of an orderly arrangement 3 points for correct volume Item 12 (6 points) a. 1 point for correct answer 1 point for explanation

The lengths of the toys are 20 cm, 30 cm, or 40 cm (2 cube lengths, 3 cube lengths, or 4 cube lengths). So the smallest possible dimensions of the box are 20 by 30 by 40 cm, if the toys can be arranged to fit. The total volume is 24,000 cm3. Students should find that there are many ways to stack the toys so that they fit into this volume.

© Houghton Mifflin Harcourt Publishing Company

b. 2 points for correct ratio 2 points for showing work

12. A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches. The vertex of the cone is directly above the center of its base. He measures halfway down the slant height and makes a cut parallel to the base. He now has a truncated cone and a cone half the height of the original. A. He expected the two parts to weigh about the same, but they don’t. Which is heavier? Why? B. Find the ratio of the weight of the small cone to that of the truncated cone. Show your work. A. The truncated cone is heavier because the bottom of the cone is wider than the top. B. The vertical cross section of the full cone is an isosceles triangle, so the height of the cone is the third side of a right triangle with a hypotenuse of 16 in. and one leg, which is the radius, of 6 in. By the Pythagorean Theorem, the height is √220 . 1 π · 6 2 · √220 = 12π √220 . The radius of the small The volume of the full cone is __ 3

―― ――

IN2_MNLESE389847_U9UC 1187

Unit 9

――

――

――

――

220 220 1 , so the volume of the small cone is __ π · 3 2 · _____ cone is 3 and its height is _____ 2 3 2 = 1.5π √220 . The volume of the truncated cone is the volume of the full cone minus the volume of the small cone: 1.5π √―― 220 __1 12π √220 - 1.5π √220 = 10.5π √220 . The ratio is _________ ―― = 7 .

Unit 9

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――

√

――

√

――

10.5π √220

1187

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math in careers

MATH IN CAREERS

model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron. Each triangular face of the octahedron is an equilateral triangle.

Jewelry Maker In this Unit Performance Task, students can see how a jewelry maker uses mathematics on the job.

a. Derive a formula for the volume of the pendant, if the side length is a. Show your work. b. The jewelry maker wants to package the pendant in a cylindrical box. What should be the smallest dimensions of the box if the pendant just fits inside, in terms of a? Explain how you determined your answer.

For more information about careers in mathematics as well as various mathematics appreciation topics, visit the American Mathematical Society http://www.ams.org

c. What is the volume of empty space inside the box? Your answer should be in terms of a, and rounded to two decimal places. Show your work. 1 a. Possible answer: For a pyramid, V = _ bh. The base is a 2. Because each 3

face is an equilateral triangle, the slant height of each triangle from the ―

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SCORING GUIDES

a√3 a 2 base, from the Pythagorean theorem, is _ + h 2 = a 2 → h = ____ . 2 2

a √― 3 ____

Task (6 points)

A cross-section of the pyramid is a triangle with sides equal to 2 and base equal to a. For this cross-section, the height of the triangle is

2 points for deriving correct formula, showing work

also equal to the height of the pyramid. Again using the Pythagorean

a√2 a √― 3 a 2 theorem, ____ = h22 + _ → h 2 = ____ . The volume of one pyramid is 2 2 2 ― ― 3 a√2 a √2 1 2 ____ _ ____ V = 3a ⋅ 2 = 6 , and the volume of the octahedrons is twice this,

or

a √― 2 ____ .

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2

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2 points for correct dimensions and explanation 2 points for correct volume, showing work

3

3

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a 2 b. Possible answer: The height of the pendant is 2 ⋅ ____ = a √2 , and 2 √

the greatest width is the diagonal of one of the bases, which is

―――

―

√a 2 + a 2 = a √2 . So the height and diameter of the cylindrical box

―

must be a√2 .

― a √― 2 πa 32 V = π ____ ⋅ a √2 = _____ , and the difference in volumes is the empty 2 2 ― ― a 3 √2 ― π - _1 ≈ 1.75a 3. πa 32 _____ _____ space, which is 2 - 3 = a 32 __ 2 3

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Unit 9

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c. Possible answer: The volume of the cylindrical box is

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Unit 9

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