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AP Calculus BC
Practice Set : Related rates
1. AP Problem : A 13 foot ladder is leaning against a wall so that the foot of the ladder is 1 foot from the wall.
A gust of wind causes the ladder to begin sliding down the wall. The motion of the ladder as it slides down
the wall is described by y 16t 2 .05t 168 where t is measured in seconds.
a. When does the top of the ladder reach the ground?
b. Determine the velocity of the end of the ladder that is resting on the ground when it is 5 feet
from the wall.
2. A reservoir in the shape of an inverted cone ( vertex down ) has a diameter of 12 feet and a height of 4 feet.
Water is flowing into the reservoir at the constant rate of 10 cubic feet per minute. At the instant when the
surface of the water is 2 feet above the vertex, what is the rate at which the water level is rising?
3. A snowball is melting at the rate of 1 cubic foot per hour. If it is always spherical :
a. how fast is the radius changing when the snowball is 18 inches in diameter?
b. how fast is the surface area changing at the same instant?
4. A water trough with trapezoidal ends is 20 feet long, 2 feet wide at the bottom, and 3 feet wide at the top,
and 2 feet tall. If water is being pumped in at a rate of 2 cubic feet per minute, how fast is the depth
increasing when the water is 1 foot deep.
5. Westway and Northline streets are straight and perpendicular to each other. A stationary police car is
located on Northline , ¼ mile from the intersection of the two streets. A sports car on Westway approaches
the intersection at the rate of 40 mph. How fast is the distance between the cars decreasing when the sports
car is 1/8 miles from the intersection?
6. AP Problem : Consider a cone with radius 6 cm and a height of 12 cm.
a. If water is leaking out a rate of 10 cubic cm per minute, how fast is the water level dropping at the
moment when the water level is 3 cm.
b. Suppose water leaks from the cone. When the water level is 6 cm., it is observed to be dropping at
the rate of 2 cm. per minute. How fast is the leak at this instant?
c. Suppose the cone is not leaking, but the water is evaporating at a rate equal to the square root
of the exposed area of water in the cone. How fast is the water level dropping when the water level
is 2 cm?
7. A balloon is released is released at a point 150 feet from an observer who is standing on level ground. If the
balloon goes straight up at a rate of 8 feet per second, answer the following:
a. What is the distance between the observer and the balloon after 8 seconds?
b. How fast is the distance between the observer and the balloon increasing when the balloon is
60 feet high?
c. How fast is the angle of elevation increasing when the balloon is 60 feet high?
8. AP Problem : A water trough is 5 feet long and its vertical crosss sections are inverted isosceles triangles
with a base of 2 feet and a height of 3 feet. Water is being siphoned out of the trough at the rate of 2 cubic
feet per minute. At any time t, let h be the depth and V be the volume in the trough.
a. Find the volume of the trough when it is full.
b. What is the rate of change in h at the instant when the trough is ¼ full by volume?
c. What is the rate of change in the area of the surface of the water at the instant when the trough is ¼
full by volume?
9. AP Problem : As shown in the figure, water is draining from a conical tank with a height of 12 feet and a
diameter 8 feet into a cylindrical tank that has a base with area 400 square feet. The depth h , in feet, of
the water in the conical tank is changing at the rate of ( h – 12 ) feet per minute.
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 ?
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.
