A13: Related Rates Practice
Please answer the following questions on a separate sheet of paper. The answers are at the bottom of the
worksheet.
1) A cube whose edge is x is contracting. When its surface area is changing at a rate equal to 6 times the rate
of change of its edge, what is the length of the edge?
2) A spherical balloon is being filled with helium at the rate of 4 cubic feet per minute. What is the rate, in
square feet per minute, at which the surface area is increasing when the volume is 288π cubic feet?
3) The height of a right circular cylinder is increasing at the rate of 3 cm per minute and the radius of the base
is decreasing at the rate of 2 cm per minute. Find the rate at which the volume of the cylinder is changing when
the height is 10 cm and the radius of the base is 4 cm.
4) Gas is being forced into a spherical balloon at the rate of 600 cubic inches per minute. How fast is the radius
of the balloon increasing when it encloses 288π cubic inches of gas?
5) A light is hung 15 feet about a straight horizontal path. If a man 6 feet tall is walking away from the light at
the rate of 5 feet/sec, how fast is his shadow lengthening?
6) The height of a cylinder is increasing at the rate of 2 inches per second. When the radius is 6 in, the volume
is increasing at 96π cubic inches per second. How fast is the lateral area increasing at that instant?
7) Gas is escaping from a spherical balloon at the rate of 2 cubic feet per minute. How fast is the surface area
shrinking when the radius is 12 feet?
8) A ladder 20 feet long leans against a house. Find the rate at which the ladder is moving downward if its foot
is 12 feet from the house and moving at a rate of 2 feet per second.
9) Water is pouring into a cylindrical bowl of height 10 feet and radius 3 feet at a rate of 5 cubic feet per
minute.
a) At what rate does the water level rise?
b) How long does it take to fill the bowl?
10) A tank is in the form of an inverted cone having an altitude of 16 meters and a base radius of 4 meters.
Water is flowing into the tank at the rate of 2 cubic meters/minute. How fast is the water level rising when the
water is 5 meters deep?
11) Water is running out of a conical funnel at the rate of 1 cubic inch per second. If the radius of the base of
the funnel is 4 inches and the altitude is 8 inches, find the rate at which the water level is dropping when it is 2
inches from the top.
12) A conical soda cup is 4 inches across the top and 5 inches deep. A soda jerk fills it at the rate of 2 cubic
inches per second. What is the vertical velocity of a bubble on the top of the liquid when the soda is halfway to
the top?
13) A point moves on the curve y  2sin( x) in such a way that when 𝑥 = 1/4, the x-coordinate is increasing
at the rate of √2 units per second. At what rate is the y-coordinate changing at that time?
14) A balloon is 200 feet off the ground and rising vertically at the constant rate of 15 feet per second. A car
passes beneath it traveling along a straight road at the constant rate of 66 feet per second (45 mph). How fast is
the distance between them changing 1 second later?
15) A baseball diamond is a square 90 feet on a side. A player is running form home to first base at the rate of
30 feet per second. When she has run halfway, at what rate is her distance from second base changing?
16) A spherical iron ball 8 inches in diameter is coated with a layer of ice of uniform thickness. If the ice melts
at the rate of 10 cubic inches per minute then
a) How fast is the thickness of the ice decreasing when it is 2 inches thick?
b) How fast is the outer surface area decreasing?
17) An air traffic controller spots 2 planes at the same altitude converging on a point as they fly at right angles
to one another. One plane is 150 miles from the point of convergence and is moving at 450 mph. The other
plane is 200 miles from the point of convergence and has a speed of 600 mph.
a) At what rate is the distance between the planes changing?
b) How much time does the traffic controller have to get one of the planes on a different flight path?
Answers: 1) ½ unit 2) 4/3 sq ft per min 3) -112π cubic cm per min 4) 25/6π in per min
5) 10/3 ft per sec 6) 28π sq in per sec 7) -1/3 sq ft per min 8) -1.5 ft per sec
9) (a) 5/9π ft per min (b) 18π min 10) 32/25π m per sec 11) -1/9π in per sec 12) 2/π in per sec
13) 2π units per sec 14) 33.69 ft per sec 15) 13.4 ft per sec 16) (a) -5/72π cu in per min
(b) -10/3 sq in per sec 17) (a) 750 mph (b) 20 min