A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 50 cm/s. Find the rate at which the area within the circle is increasing after 3 seconds

We are given the rate of change of radius of the circle.

dr/dt = 50 cm/s

Area of circle = πr²

dA/dr = 2πr

dA/dt = dA/dr x dr/dt

= 2πr x 50

= 100πr

Assuming the radius is 0 at time 0, after 3 seconds, the radius:

50 x 3 = 150 cm

Area increase rate:

100 x 150 x π

= 47,123.9 cm²/s