132 Chapter 2: Differentiation
2.6 Related Rates
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Assume that x and y are both differentiable functions of t . Find
for the equation .
a.
b.
c.
d.
e.
____ 2. Assume that x and y are both differentiable functions of t. Find
for the equation .
a.
b.
c.
d.
e.
____ 3. A point is moving along the graph of the function such that
centimeters per second. Find when x = .
a.
b.
2.6 Related Rates 133
c.
d.
e.
____ 4. A point is moving along the graph of the function such that =
centimeters per second. Find when .
a.
b.
c.
d.
e.
____ 5. The radius, r, of a circle is decreasing at a rate of centimeters per minute.
Find the rate of change of area, A, when the radius is .
a. sq cm/min
b. sq cm/min
c. sq cm/min
d. sq cm/min
e. sq cm/min
134 Chapter 2: Differentiation
____ 6. The radius r of a sphere is increasing at a rate of inches per minute. Find the rate of
change of the volume when r = inches.
a.
b.
c.
d.
e.
____ 7. A spherical balloon is inflated with gas at the rate of cubic centimeters per
minute. How fast is the radius of the balloon increasing at the instant the radius is centimeters?
a.
b.
c.
d.
e.
____ 8. All edges of a cube are expanding at a rate of centimeters per second. How fast is
the volume changing when each edge is centimeters?
a.
b.
c.
d.
e.
2.6 Related Rates 135
____ 9. A conical tank (with vertex down) is feet across the top and feet deep. If water
is flowing into the tank at a rate of cubic feet per minute, find the rate of change of the depth of the
water when the water is feet deep.
a.
b.
c.
d.
e.
____ 10. A ladder feet long is leaning against the wall of a house (see figure). The base of
the ladder is pulled away from the wall at a rate of feet per second. How fast is the top of the ladder
moving down the wall when its base is feet from the wall? Round your answer to two decimal
places.
a. ft/sec
b. ft/sec
c. ft/sec
d. ft/sec
e. ft/sec
136 Chapter 2: Differentiation
____ 11. A ladder feet long is leaning against the wall of a house (see figure). The base of
the ladder is pulled away from the wall at a rate of feet per second. Consider the triangle formed by
the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is
changed when the base of the ladder is feet from the wall. Round your answer to two decimal
places.
a.
b.
c.
d.
e.
____ 12. A ladder feet long is leaning against the wall of a house (see figure). The base of
the ladder is pulled away from the wall at a rate of feet per second. Find the rate at which the angle
between the ladder and the wall of the house is changing when the base of the ladder is feet from
the wall. Round your answer to three decimal places.
2.6 Related Rates 137
a. rad/sec
b. rad/sec
c. rad/sec
d. rad/sec
e. rad/sec
____ 13. A man 6 feet tall walks at a rate of feet per second away from a light that is 15 feet
above the ground (see figure). When he is feet from the base of the light, at what rate is the tip of
his shadow moving?
a. ft/sec
b. ft/sec
c. ft/sec
d. ft/sec
e. ft/sec
138 Chapter 2: Differentiation
____ 14. A man 6 feet tall walks at a rate of feet per second away from a light that is 15 feet
above the ground (see figure). When he is feet from the base of the light, at what rate is the length
of his shadow changing?
a. ft/sec
b. ft/sec
c. ft/sec
d. ft/sec
e. ft/sec
2.6 Related Rates 139
____ 15. A man feet tall walks at a rate of ft per second away from a light that is ft
above the ground (see figure). When he is ft from the base of the light, find the rate.at which the tip
of his shadow is moving.
a. ft per minute
b. ft per minute
c. ft per minute
d. ft per minute
e. ft per minute
____ 16. An airplane is flying in still air with an airspeed of miles per hour. If it is
climbing at an angle of , find the rate at which it is gaining altitude. Round your answer to four
decimal places.
a.
b.
c.
d.
e.
140 Chapter 2: Differentiation
2.6 Related Rates
Answer Section