2.17 Related Rates

Summary

One of the applications of mathematical modeling with calculus involves related-rates word problems. Suppose we have an equation that involves two or more quantities that are changing as functions of time. Then differentiating the equation implicitly with respect to time gives an equation that involves the rates of change of these quantities. By relating the rates in this way, we often can answer interesting questions about the model that we use to specify the original problem.

By the end of your studying, you should know:

How to set up and solve related rates word problems.

On-screen applet instructions: The slider controls the position of the runner. The applet displays the length of the runner's shadow s as a function of the runner's position x.

Examples

Suppose that an inflating balloon is spherical in shape, and its radius is changing at the rate of 3 centimeters per second. At what rate is the volume changing when the radius is 10 centimeters?

A baseball diamond is 90 feet square, and the pitcher's mound is at the center of the square. If a pitcher throws a baseball at 100 miles per hour, how fast is the distance between the ball and first base changing as the ball crosses home plate?

A ladder 10 feet long is resting against a wall. If the bottom of the ladder is sliding away from the wall at a rate of 1 foot per second, how fast is the top of the ladder moving down when the bottom of the ladder is 8 feet from the wall?