Answer:
282.743 square feet per second
Explanation:
We are told that a rock is thrown into a still pond, and that circular ripples move outward from the point of impact of the rock.
We want to find the rate at which the circular area formed by a ripple is changing when its radius is defined.
Let A be the area of the circle (in ft²).
Let r be the radius of the circle (in ft).
Let t be the time (in seconds).
Therefore, to find the rate of change of area A (with respect to time t), we need to find the equation for dA/dt.

Given the circular ripples move outward from the point of impact of the rock so that the radius of the circle formed by a ripple increases at the rate of 3 ft/s, then:

The formula for the area of a circle is A = πr².
Find dA/dr by differentiating A with respect to r:


Now we have expressions for dr/dt and dA/dr, we can multiply them to find the equation for dA/dt:



To calculate how fast the circular area is changing when the radius is 15 feet, substitute r = 15 into the found equation for dA/dt:




Therefore, when the radius is 15 feet, the area is changing at approximately 282.743 square feet per second (rounded to the nearest thousandth).