Extension: Exploring Similar Solids CK12 Editor
Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required)
To access a customizable version of this book, as well as other interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform®. Copyright © 2012 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution/NonCommercial/Share Alike 3.0 Unported (CC BY-NC-SA) License (http://creativecommons.org/licenses/by-nc-sa/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/terms. Printed: November 30, 2012
AUTHOR CK12 Editor
www.ck12.org
C ONCEPT
Concept 1. Extension: Exploring Similar Solids
1
Extension: Exploring Similar Solids
Learning Objectives • Find the relationship between similar solids and their surface areas and volumes.
Similar Solids Recall that two shapes are similar if all the corresponding angles are congruent and the corresponding sides are proportional. Similar Solids: Two solids are similar if they are the same type of solid and their corresponding radii, heights, base lengths, widths, etc. are proportional. Example 1: Are the two rectangular prisms similar? How do you know?
Solution: Match up the corresponding heights, widths, and lengths.
small prism 3 4 5 = = = large prism 4.5 6 7.5 The congruent ratios tell us the two prisms are similar. Example 2: Determine if the two triangular pyramids similar.
1
www.ck12.org Solution: Just like Example 1, let’s match up the corresponding parts. 6 8
=
3 4
=
12 16
however,
8 12
= 32 .
These triangle pyramids are not similar.
Surface Areas of Similar Solids If two shapes are similar, then the ratio of the area is a square of the scale factor.
For example, the two rectangles are similar because their sides are in a ratio of 5:8. The area of the larger rectangle is 8(16) = 128 units2 . The area of the smaller rectangle is 5(10) = 50 units2 . Comparing the areas in a ratio, it is 50 : 128 = 25 : 64 = 52 = 82 . So, what happens with the surface areas of two similar solids? Example 3: Find the surface area of the two similar rectangular prisms.
Solution:
SAsmaller = 2(4 · 3) + 2(4 · 5) + 2(3 · 5) = 24 + 40 + 30 = 94 units2 SAlarger = 2(6 · 4.5) + 2(4.5 · 7.5) + 2(6 · 7.5) = 54 + 67.5 + 90 = 211.5 units2 Now, find the ratio of the areas. of
22 . 32
94 211.5
=
4 9
=
22 . 32
The sides are in a ratio of
4 6
= 23 , so the surface areas are in a ratio
Surface Area Ratio: If two solids are similar with a scale factor of ba , then the surface areas are in a ratio of
a 2 b .
Example 4: Two similar cylinders are below. If the ratio of the areas is 16:25, what is the height of the taller cylinder? 2
www.ck12.org
Concept 1. Extension: Exploring Similar Solids
r Solution: First, we need to take the square root of the area ratio to find the scale factor, proportion to find h.
16 = 45 . Set up a 25
4 24 = 5 h 4h = 120 h = 30 Example 5: Using the cylinders from Example 4, if the area of the smaller cylinder is 1536π cm2 , what is the area of the larger cylinder? Solution: Set up a proportion using the ratio of the areas, 16:25.
16 1536π = 25 A 16A = 38400π A = 2400π cm2
Volumes of Similar Solids Let’s look at what we know about similar solids so far.
TABLE 1.1: Ratios Scale Factor Ratio of the Surface Areas Ratio of the Volumes
a b
a 2 b
??
Units in, ft, cm, m, etc. in2 , f t 2 , cm2 , m2 , etc. in3 , f t 3 , cm3 , m3 , etc.
If the ratio of the volumes follows the pattern from above, it should be the cube of the scale factor. Example 6: Find the volume of the following rectangular prisms. Then, find the ratio of the volumes. 3
www.ck12.org
Solution:
Vsmaller = 3(4)(5) = 60 Vlarger = 4.5(6)(7.5) = 202.5 The ratio is
60 202.5 ,
which reduces to
8 27
=
23 . 33
Volume Ratio: If two solids are similar with a scale factor of ba , then the volumes are in a ratio of
a 3 b .
Example 7: Two spheres have radii in a ratio of 3:4. What is the ratio of their volumes? Solution: If we cube 3 and 4, we will have the ratio of the volumes. 33 : 43 = 27 : 64. Example 8: If the ratio of the volumes of two similar prisms is 125:8, what is the scale factor? Solution: Take the cubed root of 125 and 8 to find the scale factor. √ √ 3 3 125 : 8 = 5 : 2 Example 9: Two similar right triangle prisms are below. If the ratio of the volumes is 343:125, find the missing sides in both triangles.
Solution: The scale factor is 7:5, the cubed root. With the scale factor, we can now set up several proportions.
7 7 = 5 y y=5
7 x = 5 10 x = 14
7 35 = 5 w w = 25
72 + x 2 = z2 72 + 142 = z2 √ √ z = 245 = 7 5
7 z = 5 v √ √ 7 7 5 = →v=5 5 5 v
Example 10: The ratio of the surface areas of two similar cylinders is 16:81. What is the ratio of the volumes? Solution: First, find the scale factor. If we take the square root of both numbers, the ratio is 4:9. Now, cube this to find the ratio of the volumes, 43 : 93 = 64 : 729. 4
www.ck12.org
Concept 1. Extension: Exploring Similar Solids
Review Questions • • • •
Questions 1-4 are similar to Examples 1 and 2. Questions 5-14 are similar to Examples 3-8 and 10. Questions 15-18 are similar to Example 9. Questions 19 and 20 are similar to Example 1.
Determine if each pair of right solids are similar.
1.
2.
3.
5. 6. 7. 8. 9. 10. 11. 12. 13.
4. Are all cubes similar? Why or why not? Two prisms have a scale factor of 1:4. What is the ratio of their surface areas? Two pyramids have a scale factor of 2:7. What is the ratio of their volumes? Two spheres have radii of 5 and 9. What is the ratio of their volumes? The surface area of two similar cones is in a ratio of 64:121. What is the scale factor? The volume of two hemispheres is in a ratio of 125:1728. What is the scale factor? A cone has a volume of 15π and is similar to another larger cone. If the scale factor is 5:9, what is the volume of the larger cone? The ratio of the volumes of two similar pyramids is 8:27. What is the ratio of their total surface areas? The ratio of the volumes of two tetrahedrons is 1000:1. The smaller tetrahedron has a side of length 6 cm. 5
www.ck12.org What is the side length of the larger tetrahedron? 14. The ratio of the surface areas of two cubes is 64:225. What is the ratio of the volumes? Below are two similar square pyramids with a volume ratio of 8:27. The base lengths are equal to the heights. Use this to answer questions 15-18.
18. 19. 20. 21.
What is the scale factor? What is the ratio of the surface areas? Find h, x and y. Find the volume of both pyramids.
Use the hemispheres below to answer questions 19-20.
1. Are the two hemispheres similar? How do you know? 2. Find the ratio of the surface areas and volumes.
6
