1. EXERCISE SET 4.3
In 1 – 6, assume all variables are functions of t. Expand the derivatives as much as possible.
3 2 2 d ev d u4
1. x y 2. x y 3. 4.
dt u 3 dt sin(v)
d d
5. sin 2 u ln( v) 6. cosu 3 ln( v)
dt dt
7. A rectangle is resized by dragging one of its corners. The length L is increasing at a rate of 8
cm/sec and the width W is decreasing at a rate of 2 cm/sec. What is the rate of change of the
area A at the moment the rectangle is 10 cm by 6 cm?
8. A rectangle is resized by dragging one of its corners. The length L is decreasing at a rate of9
cm/sec and the width W is increasing at a rate of 5 cm/sec. What is the rate of change of the
area A at the moment the rectangle is 12 cm by 4 cm?
9. Two cars travel on roads intersecting at an angle of 30 . Car A is going away from the
intersection at 25 m/s and car B is going toward the intersection at 20 m/s. At one instant, A is
10 m away from the intersection when B is 30 m from the intersection. How fast is the distance
between the cars changing at this instant?
10. Two cars travel on roads intersecting at an angle of 60 . Car A is going toward the
intersection at 40 ft/s and car B is going away from the intersection at 30 ft/s. At one instant, A
is 20 ft away from the intersection when B is 15 ft from the intersection. How fast is the
distance between the cars changing at this instant?
11. A bomber is flying horizontally eastward at 300 mi/hr, 6 miles above the ground. The flight
path passes directly over the target. How fast is the distance between the bomber and the
target changing when they are 10 miles apart?

2. 12. Two airplanes fly northward at the same altitude. The parallel courses are 400 miles apart.
Plane A goes 400 mi/hr and plane B goes 300 mi/hr. At noon, they are 400 mi apart. How fast
is the distance between them changing at 3 pm?
13. Water is pumped into a tank in the shape of a vertical cone, vertex down, at the rate of 4
m3/s. The tank has a diameter of 8 m and a depth of 6 m. Find the rate of change of the radius
of the water surface when the water is 3 m deep.
14. The tank in #13 is full of water. Then water is pumped out at the rate of 2 m 3/s. Find the
rate of change of the radius of the water surface when the water is 4 m deep.
15. A ladder 25 ft long rests against a house. If the lower end of the ladder slips along the
ground at the rate of 3 ft/s away from the house, how fast is the upper end of the ladder
coming down when it is 15 ft above the ground?
16. For the situation of the ladder in problem #15, what is the rate of change of the angle
between the ladder and the ground?
17. A stone is dropped into a still lake sending out a circular wave. The radius of the wave
increases at the rate of 6 ft/s. How fast is the area of the circle increasing after 4 seconds?
18. Air is escaping from a spherical balloon at the rate of 3 ft3/min. How fast is the surface area
shrinking when the radius is 12 ft?
19. A water trough 5 m long has an equilateral triangular cross section. The distance across the
top is 2 m. If water is running into the trough at the rate of 0.2 m 3/s, at what rate is the water
rising in the trough when it is two-thirds full?
20. At what rate is the cross-sectional area of the water described in problem #19 increasing?