Related Rates Problems
1. If V is the volume of the cube with edge length x and the cube expands as time passes. Find dV /dt in terms of
dx/dt. If the length of the edge is increasing at a constant speed o f 1 cm/s, how fast is the volume changing when
the edge length is 20 cm?
2. At 7:00 A.M. a truck is 60 miles due north of a car. The truck is traveling south at a constant speed of 40 mph,
while the car is traveling east at 60 mph. How fast is the distance between the car and the truck changing at 7:30
A.M.?
3. A person is sitting on a bench in a park and watching a balloon rising up in the air 100 m away from him. The
balloon is rising at a constant speed of 5 m/sec . A person moves his head in order to keep balloon in sight. How
fast does the person move his head when the balloon is at the height of 50 m?
4. Sand is being dumped into a conical pile whose height is half the radius of its base. Suppose sand is being
dumped at a rate 5 cubic meters per minute.
(a) How fast is the height of the pile increasing when it is 9 meters high?
(b) How fast is the area of the base increasing at this moment?
(c) How fast is the circumference of the base increasing at this moment?
(d) Will the height be increasing more slowly, more rapidly or at steady pace as time goes on?
5. A lighthouse is located 3 kilometers away from a long, straight beach wall. The beacon of light is rotating steadily
at a rate of 1.5 revolutions per minute.
(a) A lone soul is sitting on the beach wall 5 kilometers from the lighthouse, staring into the sea and
contemplating the universe. At what rate is the ray of light moving along the beach wall when it
passes the thinker?
(b) At what point along the beach wall is the beam moving most slowly?