Free Printable Related Rates Review

1.

A snowball spherical in shape is melting at a rate of $0.2\ \frac{\text{cm}^3}{\text{min}}$ . What is the rate of change of the radius of the snowball when the volume of the snowball is $36\pi\ \text{cm}^3$ ?

a)

$-0.628\text{}\frac{\text{cm}}{\text{min}}$

b)

$-7.2\frac{\text{cm}^3}{\text{min}}$

c)

$-3\frac{\text{cm}^3}{\text{min}}$

d)

$-0.00177\frac{\text{cm}}{\text{min}}$

2.

A person stands $30$ feet from point $P$ and watches a balloon rise vertically from that point, as shown in the figure. The balloon is rising at a constant rate of $2$ feet per second. What is the rate of change, in radians per second, of angle $\theta$ at the instant when the balloon is $40$ feet above point $P$ ?

a)

$\frac{3}{100}$

b)

$\frac{3}{125}$

c)

$\frac{1}{12}$

d)

$\frac{5}{27}$

3.

A block of ice in the shape of a cube originally having volume 1000 cm³ is melting in such a way that the length of each of its edges is decreasing at the rate of 1 cm/hr. At what rate is its surface area decreasing at the time its volume is 27 cm³? Assume that the block of ice maintains its cubical shape.

a)

dA/dt = -120 cm²/hr

b)

dV/dt = -120 cm³/hr

c)

dA/dt = 10 cm²/hr

d)

dS/dt = 10 cm²/hr

4.

A certain medical procedure requires that a balloon be inserted into the stomach and then inflated. Model the shape of the balloon by a sphere of radius r. If r is increasing at the rate of 0.3 cm/min, how fast is the volume changing when the radius is 4 cm?

a)

15.08 cm³/min

b)

268.08 cm³/min

c)

60.32 cm³/min

d)

6.03 cm³/min

5.

Hospital officials estimate that approximately N(p) = p² + 5p + 900 people will seek treatment in the emergency room each year if the population of the community is p thousand. The population is currently 20,000 and is growing at the rate of 1,200 per year. At what rate is the number of people seeking emergency room treatment increasing?

a)

48 million people per year

b)

21,200 people per year

c)

3000 people per year

d)

54 people per year

6.

A tank is in the form of an inverted cone having an altitude of 10 ft and a radius of 5 feet. Water is flowing into the tank at the rate of 1 ft³/min. How fast is the water level rising when the water is 3 ft deep?

a)

3.4 ft/min

b)

.14 ft/min

c)

1 ft/min

d)

1/2 ft/min

7.

Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The radius of the spill increases at a rate of 5 m/min. How fast is the area of the spill increasing when the radius is 5 m?

a)

50π m²/min

b)

47π m²/min

c)

52π m²/min

d)

40π m²/min

8.

A spherical snowball melts so that its radius decreases at a rate of 4 in/sec. At what rate is the volume of the snowball changing when the radius is 4 in?

a)

-262π in³/sec

b)

-247π in³/sec

c)

-256π in³/sec

d)

-263π in³/sec

9.

A 5 ft ladder is leaning against a wall and sliding towards the floor. The top of the ladder is sliding down the wall at a rate of 2 ft/sec. How fast is the base of the ladder sliding away from the wall when the base of the ladder is 3 ft from the wall?

a)

4/3 ft/sec

b)

8/7 ft/sec

c)

1 ft/sec

d)

8/3 ft/sec

10.

An observer stands 2400 ft away from a launch pad to observe a rocket launch. The rocket blasts off and maintains a velocity of 200 ft/sec. Assume the scenario can be modeled as a right triangle. How fast is the observer to rocket distance changing when the rocket is 700 ft from the ground?

a)

56 ft/sec

b)

57 ft/sec

c)

52 ft/sec

d)

61 ft/sec

11.

A water tank, shaped like an inverted circular cone, has a base radius of 6 ft and a height of 9 ft. The tank is completely full and needs to be drained. The valve is opened and the water begins to decrease at a rate of 2 ft³/sec. How fast is the height of the water changing when the water is 2 ft deep?

a)

-9/(8pi) ft/sec

b)

9/(8pi) ft/sec

c)

-8/(9pi) ft/sec

d)

8/(9pi) f/tsec

12.

Devin set up a toy rocket. For safety, he stands 6 meters from the rocket. He sets off the rocket and it heads straight up at a constant rate of 4 m/s. How fast is the distance between the rocket and Devin changing after 2s?

a)

-2.5 m/s

b)

2.5 m/s

c)

3.2 m/s

d)

-3.2 m/s

13.

A pebble is thrown into a pond forming a ripple whose radius increases at a rate of 4 in/sec. How fast is the area of the ripple changing when the radius is 1 foot?

a)

$2\pi\ \frac{in^2}{\sec}$

b)

$24\pi\ \frac{in^2}{\sec}$

c)

$\frac{2\pi}{3}\ \frac{ft^2}{\sec}$

d)

$2\pi\ \frac{ft^2}{\sec}$

e)

$8\pi\ \frac{ft^2}{\sec}$

14.

A 20-foot ladder leans against the wall of a building. The ladder starts sliding down the wall so that the top of the ladder moves down at the rate of 0.5 ft/sec. How fast is the foot of the ladder moving away from the wall when the foot of the ladder is 12 feet from the wall?

a)

0.5 ft/sec

b)

$\frac{5}{8}$ ft/sec

c)

$\frac{2}{3}$ ft/sec

d)

$\frac{4}{3}$ ft/sec

e)

$\frac{8}{3}$ ft/sec

15.

A spherical balloon is filled with air at $8\ \frac{in^3}{\sec}$ . How fast is the diameter of the balloon increasing when the volume of the balloon is $36\pi\ in^3$ ?

a)

$\frac{4}{9\pi}\ \frac{in}{\sec}$

b)

$\frac{2}{3\pi}\ \frac{in}{\sec}$

c)

$\frac{2}{9\pi}\ \frac{in}{\sec}$

d)

$\frac{8}{27\pi}\ \frac{in}{\sec}$

e)

$\frac{2}{27\pi}\ \frac{in}{\sec}$

Related Rates Review

Total questions: 15

Worksheet time: 3hrs 46mins

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