Question

The radius of a circle is increasing uniformly at the rate of $3cm/s$. Find the rate at which the area of the circle is increasing when the radius is $10cm$.

Answer 1

The area of a circle $\left(A\right)$ with radius $\left(r\right)$ is given by,
$A=\pi r^2$
$\frac{dA}{dt}=\frac{d}{dt}\left(\pi r^2\right)\cdot \frac{dr}{dt}=2\pi r\frac{dr}{dt}$, [By chain rule]
It is given that,
$\frac{dr}{dt}=3cm/s$.
$\therefore \frac{dA}{dt}=2\pi r\left(3\right)=6\pi r$
Thus, when $r=10cm$,
$\therefore \frac{dA}{dt}=6\pi \left(10\right)=60\pi c{m}^2/s$.