Related Rates: Master the Art of Solving Change Problems

Question 1

A spherical balloon is being inflated with air at a constant rate of 10 cubic centimeters per second. Find the rate of change of the balloon's radius when its radius is 5 centimeters.

Accepted Answer

0.4 cm/s

Answer

0.2 cm/s

Answer

1 cm/s

Answer

2 cm/s

Question 2

A spherical snowball is melting at a constant rate of 2 cubic inches per minute. If the snowball's radius is currently 3 inches, what is the rate of change of its surface area with respect to time?

Accepted Answer

-16π square inches per minute

Answer

2π square inches per minute

Answer

2 cubic inches per minute

Answer

-2 cubic inches per minute

Question 3

A spherical balloon is being deflated at a constant rate of 100 cubic centimeters per second. Find the rate of change of the balloon's surface area when its radius is 5 centimeters.

Accepted Answer

-100 square centimeters per second

Answer

50 square centimeters per second

Answer

150 square centimeters per second

Answer

200 square centimeters per second

Question 4

A ladder 10 meters long leans against a vertical wall. The base of the ladder is sliding away from the wall at a rate of 2 meters per second. How fast is the top of the ladder sliding down the wall when the base of the ladder is 6 meters from the wall?

Accepted Answer

3.33

Answer

4.16

Answer

2.50

Answer

5.00

Question 5

A car is traveling along a straight road at a speed of 90 kilometers per hour. If the driver applies the brakes and decelerates at a constant rate of 5 meters per second squared, how long will it take the car to come to a complete stop?

Accepted Answer

10

Answer

15

Answer

20

Answer

25

Question 6

A rectangular garden is 5 meters long and 3 meters wide. If the length is increasing at a rate of 2 meters per hour and the width is increasing at a rate of 1 meter per hour, what is the rate of change of the area of the garden after 2 hours?

Accepted Answer

11

Answer

13

Answer

9

Answer

15

Question 7

A spherical balloon is being inflated at a rate of 6 cubic centimeters per second. At the instant the radius is 5 centimeters, what is the rate of change of the surface area of the balloon?

Accepted Answer

6π cm²/s

Answer

10π cm²/s

Answer

3π cm²/s

Question 8

A ladder 10 feet long is leaning against a wall. The base of the ladder is sliding away from the wall at a rate of 2 feet per second. How fast is the top of the ladder sliding down the wall when the base is 6 feet from the wall?

Accepted Answer

20/3 ft/s

Answer

10/3 ft/s

Answer

15/2 ft/s

Answer

5 ft/s

Question 9

A conical tank has a height of 10 feet and a radius of 5 feet at the bottom. Water is flowing into the tank at a rate of 10 cubic feet per minute. How fast is the water level rising when the water is 6 feet deep?

Accepted Answer

0.12 ft/min

Answer

0.06 ft/min

Answer

0.24 ft/min

Answer

0.48 ft/min

Question 10

A rectangular garden is 6 feet long and 4 feet wide. The width is increasing at a rate of 1 foot per day. At what rate is the area of the garden increasing when the width is 5 feet?

Accepted Answer

6 ft²/day

Answer

9 ft²/day

Answer

3 ft²/day

Question 11

A cylindrical tank has a radius of 3 feet and a height of 5 feet. Water is flowing out of the tank at a rate of 2 cubic feet per minute. How fast is the water level dropping when the water is 2 feet deep?

Accepted Answer

0.16 ft/min

Answer

0.32 ft/min

Answer

0.08 ft/min

Answer

0.64 ft/min

Question 12

A spherical snowball is melting at a rate of 1 cubic inch per minute. What is the rate of change of the surface area of the snowball when the radius is 2 inches?

Accepted Answer

-4π in²/min

Answer

-2π in²/min

Answer

-8π in²/min

Answer

-16π in²/min

Question 13

A helicopter is hovering at a height of 500 feet above the ground. A ball is dropped from the helicopter and falls at a constant rate of 32 feet per second. What is the rate of change of the distance between the helicopter and the ball 2 seconds after the ball is dropped?

Accepted Answer

-32 ft/s

Answer

-16 ft/s

Answer

-48 ft/s

Question 14

A car is traveling along a straight road at a speed of 60 miles per hour. The car is decelerating at a constant rate of 10 miles per hour per second. What is the rate of change of the distance traveled by the car 5 seconds after it starts decelerating?

Accepted Answer

-50 mph

Answer

-100 mph

Answer

-75 mph

Answer

-25 mph

Question 15

A ladder 15 feet long is leaning against a wall. The base of the ladder is sliding away from the wall at a rate of 2 feet per second. How fast is the top of the ladder sliding down the wall when the base is 9 feet from the wall?

Accepted Answer

40/9 ft/s

Answer

20/9 ft/s

Answer

30/9 ft/s

Question 16

A spherical balloon is being inflated at a rate of 12 cubic centimeters per second. At the instant the radius is 4 centimeters, what is the rate of change of the volume of the balloon?

Accepted Answer

48π cm³/s

Answer

24π cm³/s

Answer

96π cm³/s

Answer

72π cm³/s