Question

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

Answer 1

The area of a circle $\left(A\right)$ with radius $\left(r\right)$ is given by $A=\pi r^2$.

Therefore, the rate of change of area $\left(A\right)$ with respect to time $\left(t\right)$ is given by,

$\frac{dA}{dt}=\frac{d}{dt}\left(\pi r^2\right)=\frac{d}{dr}\left(\pi r^2\right)\frac{dr}{dt}=2\pi r\frac{dr}{dt},$ [By chain rule]

It is given that $\frac{dr}{dt}=5cm$

Thus, when $r=8cm,$

$\frac{dA}{dt}=2\pi \left(8\right)\left(5\right)=80\pi c{m}^2/s$