MATHEMATICS 201-103-RE
12 – RELATED RATES
PHILIP FOTH
1. The area of a square is increasing at a rate of 36 cm2/s. At what rate is the side increasing
when the area of the square is 81 cm2?
2. The length of a rectangle is increasing at a rate of 2 cm/s, while the area is decreasing at a rate
of 12 cm2/s. At what rate is the width changing when the area is 32 cm2 and the length is 8 cm?
3. The height of a triangle is increasing at a rate of 3 m/s, while the base is decreasing at a rate
of 4 m/s. How fast is the area changing when the base is 5 m, and the height is 6m? Is it
increasing or decreasing?
4. The circumference of a circle is increasing at a rate of 15 ft/s. How fast is the area increasing
when the radius equals 5 ft?
5. The volume of a cube is increasing at a rate of 36 m3/min. How fast is its side changing when
the volume equals 216 m3?
6. The radius of a spherical balloon is decreasing at a rate of 1.5 cm/s. How fast is its volume
decreasing when the radius equals 9 cm?
7. The surface area of a spherical cell is increasing at a rate of 44 mm2/min. How fast is the
volume of the cell increasing when the radius equals 3 mm?
8. A cylindrical tank is being filled with water at a rate of 12 ft3/min. How fast is the height of
the water increasing if the diameter of the tank equals 4 ft?
9. At noon, ship A is 120 km east of ship B. Ship A is sailing west at 30 km/h and ship B is
sailing north at 40 km/h. How fast is the distance between the ships changing at 2:00 pm?
10. A street light pole has a height of 12 ft. A man 6 ft tall walks away from the pole at a speed
of 4 ft/s. How fast is the tip of his shadow moving when he is 24 ft away from the pole?
11. A spotlight on the ground shines on a wall 12 meters away. A child 1 meter tall walks from
the spotlight towards the wall at a speed of 0.8 m/s. How fast is the height of his shadow
decreasing when he is 8 meters away from the wall?
12. Two cars start moving from the same point. One goes north at 20 m/s, and the other travels
east at 15 m/s. How fast is the distance between the cars changing 10 seconds later? How about
20 seconds later?
13. A boat is pulled towards a dock by a rope attached to a pulley that is 1.5 meters higher than
the boat. If the rope is pulled with the speed of 1.2 m/s, how fast is the boat approaching the
dock, when it is 3.6 meters away from the dock?
14. Wet cement is dumped from a pipe at a rate of 12 m3/hour and it forms a conical pile whose
base diameter is always four times bigger than its height. How fast is the height of the pile
increasing when the base diameter is equal to 4 meters?
15. A tank has the shape of inverted cone, whose aperture angle at the vertex
equals 60o. The tank is filled with water, which is leaking from the vertex at
such a rate that the level of the water decreases by 5 cm/min. Find that rate
when the height of the water level was 60 centimeters.
16. A ladder 10 ft long leans against a vertical wall. It begins to slide, so the top moves down the
wall and the bottom on the ground slides away from the wall at a rate of 1.2 ft/s. How fast is the
top sliding at the moment when the bottom of the ladder is 8 ft away from the wall?
17. In the setting of the previous problem, how fast is the angle formed by the ladder and the
ground changing at the moment when the bottom of the ladder is 8 ft away from the wall?
18. A lighthouse is located on an island 3 km away from the nearest point P on a straight
shoreline and it makes 6 revolutions per minute. How fast is the beam of light moving along the
shoreline, when it is 2 km away from P ?
19. A swimming pool is 10 m wide, 12 m long, 2 m deep at the deepest point. A cross-section is
shown in the figure. The pool is being filled with water at a rate of 1.5 m3/min. How fast is the
water level rising when its depth is 1 m ?
20. A plane is flying horizontally at an altitude of 3 km and a speed of 600 km/h. It passes
directly over a tracking telescope. How fast is the distance between the telescope and the plane
changing at the moment when the plane is at a distance of 5 km from the telescope?
21. In the setting of the previous problem, how fast is the angle of elevation changing at the
moment when the plane is away at a distance of 5 km from the telescope? Give your answer in
terms of radians per minute.
ANSWERS
1. 2 cm/s
2. −2.5 cm/s
7. 66 mm3/min
13. 1.3 m/s
3. −4.5 m2/s
8. 3 /  ft/min
4. 75 ft2/s
9. 14 km/h
10. 8 ft/s
14. 3 /  m/h 15. 6000  cm3/min
18. 52  km/min
19.
3
m/min
160
5. 1/3 m/min
20. 480 km/h
6. 486  cm3/s
11 0.6 m/s
16. −1.6 ft/s
12. 25 m/s
17. −0.2 rad/s
21. 1.2 rad/min