### Related Rates Greatest Hits

#### by Professor DeTurck

- A light is at the top of a 16-ft pole. A boy 5 ft tall walks
away from the pole at a rate of 4 ft/sec. At what rate is the tip
of his shadow moving when he is 18 ft from the pole? At what rate
is the length of his shadow increasing?

- A man on a dock is pulling in a boat by means of a rope
attached to the bow of the boat 1 ft above the water level and
passing through a simple pulley located on the dock 8 ft above
water level. If he pulls in the rope at a rate of 2 ft/sec, how
fast is the boat approaching the dock when the bow of the boat is
25 ft from a point on the water directly below the pulley?

- A weather balloon is rising vertically at a rate of 2 ft/sec.
An observer is situatied 100 yds from a point on the ground
directly below the balloon. At what rate is the distance between
the balloon and the observer changing when the altitude of the
balloon is 500 ft?

- The ends of a water trough 8 ft long are equilateral triangles
whose sides are 2 ft long. If water is being pumped into the
trough at a rate of 5 cu ft/min, find the rate at which the water
level is rising when the depth is 8 in.

- Gas is escaping from a spherical balloon at a rate of 10 cu
ft/hr. At what rate is the radius chaing when the volume is 400 cu
ft?

- An airplane, flying at a constant speed of 360 mph and
climbing at an angle of 45 degrees, passes over a point P on the
ground at an altitude of 10,560 ft. Find the rate at which its
distance from P is changing one minute later.