Related Rates: Master the Dynamics of Changing Quantities

Question 1

A spherical balloon is being inflated at a rate of 10 cubic centimeters per second.  What is the rate of change of the balloon's radius when the radius is 5 centimeters?

Accepted Answer

2 cm/s

Answer

5 cm/s

Answer

10 cm/s

Answer

25 cm/s

Question 2

A spherical snowball is melting at a constant rate of dV/dt = -2 cm³/min. What is the rate of change of the snowball's surface area (dS/dt) when the radius is 5 cm?

Accepted Answer

dS/dt = 4π cm²/min

Answer

dS/dt = 2π cm²/min

Answer

dS/dt = 6π cm²/min

Answer

dS/dt = 8π cm²/min

Question 3

A spherical balloon is being inflated at a rate of 5 cubic centimeters per second. What is the rate at which the radius of the balloon is increasing when the radius is 10 centimeters?

Accepted Answer

1/(10π) centimeters per second

Answer

1/(5π) centimeters per second

Answer

10/(π) centimeters per second

Question 4

A 10-foot ladder is leaning against a vertical wall. The base of the ladder is pulled 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

8/3 feet per second

Answer

2 feet per second

Answer

3/8 feet per second

Question 5

A spherical balloon is deflating at a constant rate of 5 cubic centimeters per second. How fast is the radius of the balloon decreasing when the radius is 10 centimeters?

Accepted Answer

1/30 centimeters per second

Answer

5 centimeters per second

Answer

10 centimeters per second

Answer

1/10 centimeters per second

Question 6

A conical tank has a radius of 8 feet at the base and a height of 12 feet. Water flows out of the tank at a constant rate of 15 cubic feet per minute. How fast is the height of the water decreasing when the water level is 6 feet deep?

Accepted Answer

5/9 feet per minute

Answer

15 feet per minute

Answer

5/18 feet per minute

Answer

12 feet per minute

Question 7

A 10-meter ladder is leaning against a vertical wall. The bottom of the ladder is pulled away from the wall at a rate of 1 meter per second. When the bottom of the ladder is 6 meters from the wall, what is the rate of change of the top of the ladder sliding down the wall?

Accepted Answer

8/3 meters per second

Answer

3/8 meters per second

Answer

1 meter per second