Related Rates

An application of the derivative

Formulas you should know

Trust me!

Formulas

Triangles :

Pythagorean Theorem

a2 +b2 =c2

Area

A=12bh

Rectangular Prisms

v =lwhSA=2lw+ 2lh+ 2wh

Circles

A =πr2

C =2πr

Formulas

Cylinders

V =πr2hLSA=2πrhSA=2πrh+ 2πr2

Spheres

V =43πr3

SA=4πr2

Right Circular Cone

If one of these is on the test, I’ll give you the formula!

V =πr2h

3

LSA=πr r2 +h2

SA=πr r2 +h2 +πr2

Example #1

A circular pool of water is expanding at the rate of 16π in2/sec. At what rate is the radius expanding when the radius is 4 inches?

Example #2

A 25-foot ladder is leaning against a wall and sliding toward the floor. If the foot of the ladder is sliding away from the base of the wall at a rate of 15 ft/sec, how fast is the top of the ladder sliding down the wall when the top of the ladder is 7 feet from the ground?

Example #3

A spherical balloon is expanding at a rate of 60π in3/sec. How fast is the surface area of the balloon expanding when the radius of the balloon is 4 inches?

Example #4

An underground conical tank, standing on its vertex, is being filled with water at a rate of 18π ft3/min. If the tank has a height of 30 feet and a radius of 15 feet, how fast is the water level rising when the water is 12 feet deep?

Example #5

A circle is increasing in area at a rate of 16π in2/sec. How fast is the radius increasing when the radius is 2 inches?

Example #6

A rocket is rising vertically at a rate of 5400 mph. An observer on the ground is standing 20 mi from the launch point. How fast (in radians/hr) is the angle of elevation b/w the ground and the observers line of sight is the rocket increasing when the rocket is at an elevation of 40 miles?

Example #7

Water is being drained out of a conical tank. Suppose the height is changing at a rate of -.2 ft/min and the radius is changing at a rate of -.1 ft/min. What is the rate of change in the volume when the radius is 1 and the height is 2?

Example #8

A pebble is dropped into a calm pond causing ripples in the form of circles. The radius of the outer ripple is increasing at a constant rate of 1 ft/sec. When the radius is 4 feet, at what rate is the area of the disturbed water changing?

Example #9

Air is being pumped into a spherical balloon at a rate of 4.5 cubic in/min. Find the rate of change of the radius when the radius is 2 inches.

Example #10

An airplane is flying on a flight path that will take it directly over a radar station. If the distance between the radar and the plane is decreasing at a rate of 400 mi/hr when that distance is 10 miles, what is the speed of the plane? The plane is traveling at an altitude of 6 miles.

Example: From text #11(will have variables in answer)

Find the rate of change of the distance between the origin and a moving point on the graph of y=x2+1 if dx/dt = 2 cm/sec

Example #12

Suppose a spherical balloon is inflated at the rate of 10 cubic cm per min. How fast is the radius of the balloon increasing when the radius is 5 cm?

Example #13

One end of a 13-ft ladder is on the floor and the other end rests against a vertical wall. If the bottom of the ladder is drawn away from the wall at 3 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 5 feet from the wall?

Example #14

Water is poured into a conical paper cup at the rate of 2/3 cubic inches per sec. If the cup is 6 in tall and the top of the cup has a radius of 2 in, how fast does the water level rise when the water is 4 in deep?

Example #15

Pat walks at the rate of 5 ft/sec toward a street light whose lamp is 20 ft above the base of the light. If Pat is 6 ft tall, determine the rate of change of the length of Pat’s shadow at the moment when Pat is 24 feet from the base of the lamppost.

Example #16