AP Calculus AB AP Review #7 Homework _____ 1.
Related Rates
Name__________________________________ Date______________________ Period _______
The volume of a cone of radius r and height h is given by V =
1 2 πr h . If the radius and the 3
1 centimeter per second, at what rate, in cubic 2 centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters? 85AB
height both increase at a constant rate of
A.
_____ 2.
1 π 2
16 9
E. 108π
B. 57.88
C. 59.20
D. 60.00
E. 67.40
B.
4 3
C.
3 4
D.
4 9
E.
4 27
If the rate of change of f(x) at x = c is twice the rate of change of f(x) at x = 4, and f ( x ) = 2ln x , then c is A. 1
_____ 5.
D. 54π
A person 2 meters tall walks directly away from a streetlight that is 8 meters above the ground. If the person is walking at a constant rate and her shadow is lengthening at a rate of 4/9 meters per second, at what rate in meters per second is the person walking? A.
_____ 4.
C. 24π
A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the crossing and watches an eastbound train traveling at 60 meters per second. At how many meters per second is the train moving away from the observer 4 seconds after it passes through the intersection? A. 57.60
_____ 3.
B. 10π
B. 2
C. 4
D. 8
E. 16
The radius of a sphere is decreasing at a constant rate of 0.04 centimeters per second. At the time that the radius is 10 centimeters, find the rate of change ( in cubic centimeters per second) of the volume. (Note: The volume of a sphere of radius r is given by V = A. 57.19
B. 5.03
C. -5.03
D. 50.27
4 3 π r .) 3
E. -50.27
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AP Review #7 Homework 6.
As shown in the figure below, water is draining from a conical tank with height 12 feet and diameter 8 feet into a cylindrical tank that has a base with area 400π square feet. The depth h, in feet, of the water in the conical tank is changing at the rate (h – 2) feet per minute. (The volume V of a cone 1 with radius r and height h is V = πr 2h .) 3 95AB, BC - #5
(a)
Write an expression for the volume of water in the conical tank as a function of h.
(b)
At what rate is the volume of water in the conical tank changing when h = 3? Indicate units of measure.
(c)
Let y be the depth, in feet, of the water in the cylindrical tank. At what rate is y changing when h = 3? Indicate units of measure.
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AP Review #7 Homework 7.
A container has the shape of an open right circular cone, as shown in the figure below. The height of the container is 10 cm and the diameter of the opening is 10 cm. Water in the container is 3 evaporating so that its depth h is changing at the constant rate of − cm/hr. 10 02AB - #5
(a)
Find the volume V of water in the container when h = 5 cm. Indicate units of measure.
(b)
Find the rate of change of the volume of water in the container, with respect to time, when h = 5 cm. Indicate units of measure.
(c)
Show that the rate of change of the volume of water in the container due to evaporation is directly proportional to the exposed surface area of the water. What is the constant of proportionality?
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AP Review #7 Homework (non calc) 07AB #5
t (minutes) r’(t) (feet per minute)
0
2
5
7
11
12
5.7
4.0
2.0
1.2
0.6
0.5
8. The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon in feet is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For 0 < t < 12, the graph of r is concave down. The table above gives selected values of the rate of change, r ‘(t), of the radius of the balloon over the time interval 0 < t < 12. The radius of the balloon is 30 feet when t = 5. (Note: The volume of a sphere of radius r is given by V =
4 3 π r .) 3
(a) Estimate the radius of the balloon when t = 5.4 using the tangent line approximation at t = 5. Is your estimate greater than or less than the true value? Give a reason for your answer.
(b) Find the rate of change of the volume of the balloon with respect to time when t = 5. Indicate units of measure.
(c) Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate
∫
12
0
r '(t ) dt . Using correct units, explain the meaning of
∫
12
0
r '(t ) dt in terms of the
radius of the balloon.
(d) Is your approximation in part (c) greater than or less than
∫
12
0
r '(t ) dt ? Give a reason for your answer.
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