Related Rates Problems: Master the Art of Calculus III

Question 1

A rectangular garden is 10 feet long and 6 feet wide. The length of the garden is increasing at a constant rate of 2 feet per minute, and the width is decreasing at a constant rate of 1 foot per minute. At what rate is the area of the garden changing when the length is 12 feet and the width is 5 feet?

Accepted Answer

7 sq ft/min

Answer

-7 sq ft/min

Answer

12 sq ft/min

Answer

-12 sq ft/min

Question 2

Two cars leave an intersection at the same time. One car travels north at 30 mph and the other travels east at 40 mph.  What is the rate of change of the distance between the cars after 2 hours?

Accepted Answer

50 mph

Answer

100 mph

Answer

70 mph

Answer

140 mph

Question 3

A plane flies horizontally at a constant altitude of 5000 meters and a speed of 600 km/h. The plane passes directly over a radar station.  What is the rate of change of the distance between the plane and the radar station 1 minute after the plane passes over the station?

Accepted Answer

599.6 km/h

Answer

600.4 km/h

Answer

600 km/h

Question 4

A particle moves along the curve y = x^2.  What is the rate of change of the distance between the particle and the point (2, 0) when x = 1 and the x-coordinate of the particle is increasing at a rate of 3 units per second?

Accepted Answer

4.243 units per second

Answer

6 units per second

Answer

3 units per second

Answer

9 units per second

Question 5

A plane flying horizontally at a speed of 500 miles per hour passes directly over a radar station. One minute later, the radar detects that the plane is 5 miles away from the station. How fast is the plane's altitude increasing at that moment?

Accepted Answer

499.95 mph

Answer

450.00 mph

Answer

499.50 mph

Answer

495.00 mph

Question 6

A water tank in the shape of a right circular cone is being filled with water at a rate of 5 cubic meters per minute. The radius of the base of the cone is 2 meters and the height is 4 meters. How fast is the water level rising when the water level is 1 meter high?

Accepted Answer

1.25 m/min

Answer

2.50 m/min

Answer

3.75 m/min

Answer

5.00 m/min

Question 7

A car is traveling along a straight road at a constant speed of 60 miles per hour. A stationary observer is 1 mile away from the road. How fast is the distance between the car and the observer increasing when the car is 2 miles away from the observer?

Accepted Answer

53.67 mph

Answer

30 mph

Answer

40 mph

Answer

60 mph

Question 8

A conical tank with a radius of 2 m and a height of 3 m is being filled with water at a rate of 5 m³/min. What is the rate of change of the water level when the water level is 1 m deep?

Accepted Answer

0.42 m/min

Answer

0.21 m/min

Answer

0.84 m/min

Answer

1.68 m/min

Question 9

The perimeter of a square is decreasing at a rate of 8 cm/s. What is the rate of change of the area of the square when the perimeter is 24 cm?

Accepted Answer

-16 cm²/s

Answer

-8 cm²/s

Answer

16 cm²/s

Answer

8 cm²/s

Question 10

The volume of a cube is decreasing at a rate of 27 cm³/s. What is the rate of change of the cube's edge length when the edge length is 6 cm?

Accepted Answer

-1.5 cm/s

Answer

0.75 cm/s

Answer

1.5 cm/s

Answer

-0.75 cm/s

Question 11

A boat is traveling east at 20 km/h while a second boat is traveling south at 15 km/h.  How fast is the distance between the boats changing when the first boat is 5 km east of the starting point and the second boat is 12 km south of the starting point?

Accepted Answer

25 km/h

Answer

35 km/h

Answer

15 km/h

Answer

10 km/h

Question 12

A shadow cast by a 6-foot-tall person is lengthening at a rate of 2 feet per second. If the person is walking away from a light source at a constant rate, how fast is the person walking?

Accepted Answer

Cannot be determined

Answer

1 ft/s

Answer

3 ft/s

Answer

6 ft/s