A stone is dropped in a quiet lake and it is observed that waves move in circles, If the radius of a…
A stone is dropped in a quiet lake and it is observed that waves move in circles, If the radius of a circular wave increases at the rate $2 \mathrm{~cm} / \mathrm{sec}$, then the rate of increase in its area at the instant when its radius is $10 \mathrm{~cm}$, is $\mathrm{cm}^2 / \mathrm{sec}$.
$40 \pi$
$80 \pi$
$10 \pi$
$20 \pi$
Solution
Given $\frac{d r}{d t}=2 \mathrm{~cm} / \mathrm{sec} r=10 \mathrm{~cm}$
We have $A=\pi r^2$
Diff. w.r.t. $\mathrm{t}$
$\begin{aligned} & \frac{d A}{d t}=2 \pi r \frac{d r}{d t} \\ & =2 \pi \times 10 \times 2 \\ & =40 \pi\end{aligned}$