Problem

Robert throws a rock into a lake, which creates a circle of ripples which moves away from the point of impact at a constant speed of 50 centimeters per second.

What is the rate of change in the area of the circle after 1 second? After 10 seconds? After t seconds?

Solution

First, we find the formula for the radius of the circle of ripples at any time t. The ripples move outward from the point where the rock entered the pond at a speed of 50 centimeters per second, so the radius of the circle is 50t centimeters. The area of the circle at time t is

The rate of change in the area of the circle can be found by taking the derivative. Since we have a formula for the area at any time t, the derivative A'(t) will give us a formula for the rate of change of the area with respect to time at any time t.

Using the power rule,

After 1 second, the rate of change in the area of the circle is

After 10 seconds,