A stone is dropped into a quiet lake and waves move in circles at the rate of 5 cm/sec. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?

The correct answer is $80 \pi$ $cm^2/sec$

Let $r$ cm be the radius of the circular wave at time $t$

Rate of increase of radius of circular wave = $\dfrac{dr}{dt}$ = $5$ cm/s
Consider $a$ be the area of circular wave.

∴ $a$ = $π r^2$

Rate of change of area = $\dfrac{da}{dt}$ = $π \dfrac{d}{dt} r^2$
= $2r π \dfrac{dr}{dt}$
= $2r π (5)$ = $10 \pi r$


At the instant when the radius of circular wave is 8 cm, $\dfrac{da}{dt}$ = $ 80 \pi$

As $\dfrac{da}{dt}$ is positive, the area of circular wave is increasing at the rate of $80 \pi$ $cm^2/sec$