2.8 Homework: Answers
1. A rectangular swimming pool (with horizontal bottom) is being drained. If its length and width are 25
feet and 20 feet and the water level is falling at the rate of 0.5 ft/min., how fast is the water draining?
250 ft3/min
2. Rain is falling into a cylindrical barrel at the rate of 20 cm3/min. If the radius of the base is 18cm, how
fast is the water rising? 0.02 cm/min
3. An oil spill is expanding in a circular pattern outwards with the radius increasing at the rate of 6.2
meters per hour. How fast is the surface area inside the circle changing when the radius is k meters?
4. An icicle is in the shape of a right circular cone. At a certain point in time the length is 15 cm. and is
increasing at the rate of 1 cm./hr., while the radius of the base is 2 cm. and is decreasing at 1/10
cm/hr. Is the volume of ice increasing or decreasing at that instant? at what rate? Decreasing by 2.09
cm3/hr
5. A car traveling north at 40 mph and a truck traveling east at 30 mph leave an intersection at the same
time. At what rate will the distance between them be changing 3 hours later? 50 mph
6. A man 6 ft tall is walking away from a lamppost at the rate of 50 ft per minute. When the man is 8 ft
from the lamppost, his shadow is 10 ft long. Find the rate at which the length of the shadow is
increasing when he is 25 ft from the lamppost. 62.5 ft/min
7. The price p (in dollars) and demand x for a product are related by 2𝑥 2 + 5𝑥𝑝 + 50𝑝2 = 80,000.
a. If the price is increasing at a rate of $2 per month when the price is $30, find the rate of change of
the demand. Decreasing by 12.72 units/month
b. If the demand is decreasing at a rate of 6 units per month when the demand is 150 units, find the
rate of change of the price. Increasing by $1.53/month
8. Each edge of a cube is increasing at a rate of 3 cm/sec. Find the rate of change each of the following
when the edges of the cube are 20 cm long.
a) surface area 720 cm2/sec
b) volume 3600 cm3/sec
9. A water trough is 10 ft long. The vertical ends are isosceles triangles, 4 feet across the top with
sloping sides of length 3 feet. Water is flowing into the trough at a rate of 2 ft3/min. Find the rate of
change of the depth of water in the trough when the depth is 1.5 feet. 0.15 ft/min
10. A water trough is 12 feet long. The vertical ends are trapezoids that are 4 feet across the top, 1 foot
deep, and 2 feet across the bottom. Water is pouring into the trough at a rate of 8 ft3/min. Find the
rate of change of the depth of water in the trough when the depth is 0.2 feet. 0.28 ft/min
11. A TV camera at the center of a circular track of radius 30 meters follows a runner around the track. At
a certain instant the camera is turning at 10 per second. How fast is the runner moving at that
instant? 5.24 m/sec
12. A hot-air balloon is rising vertically. An observer on the ground, 5000 ft from the point where the
balloon took off, is watching it rise. When the balloon reaches a height of 10,000 ft, its velocity is 15
ft/sec. How fast is the angle of elevation of the balloon changing at that instant? 0.0006 radians/sec
13. A portion of a roller coaster track has the shape of the curve y 
4
, 2  x  2. As a car moves
1 x2
along the track, its shadow is cast on the ground (the x  axis) by the sun’s rays (parallel to the y 
axes). Distance is measured in meters.
a. When the car reaches the point ( 1, 2 ) its shadow is moving to the right at 2.7 m/s. How fast is
the car gaining altitude at that instant? 5.4 m/s
b. What is the rate of change of altitude as the car passes through the point ( 0, 4)? Does the answer
depend on how fast the car is moving? It is not changing no matter how fast the car is moving.
c. The car is losing altitude at the rate of 22.2 m/s when it passes through the point (1,2). How fast is
its shadow moving then? 11.1 m/s
14. a. A hemispherical bowl with a radius of 10 cm is filled with water to a depth of ℎ cm. Find the radius
of the surface of the water as a function of ℎ. 𝑟 = √20ℎ − ℎ2 cm
b. The water level drops at a rate of 0.1 cm per hour. At what rate is the radius of the water
decreasing when the depth is 5cm? 0.0577 cm/hr
15. A baseball diamond is a square with side lengths of 90 feet. A batter hits the ball and runs toward
first base with a speed of 24 ft/s. At what rate is his distance from second base decreasing when he is
halfway to first base? 10.73 ft/s
16. A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight
toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building
decreasing when he is 4 m from the building. 0.6 m/s