A stone is tossed into a pond creating a circular ripple whose radius increases at a rate of 3 feet per second. In square feet per second, how fast is the area of the ripple increasing 15 seconds after the stone hits the water?

Question

anonymous · Answer

Givens:
$$dr/dt = 3ft/second$$
$$A = πr^2$$
$$dA/dr = 2πr$$
$$dA/dt = dA/dr * dr/dt$$
$$dA/dt = 2πr*3ft/second$$
$$dA/dt = 6πr$$
At 15 seconds, the radius of the ripple is 45 ft (assuming the ripple starts with a radius of 0 ft).  Therefore:$$dA/dt = 270 ft^2/second$$