Question

Find the rate of change of the area of a circle per second with respect to its radius $r$ when $r=5\text{cm}.$

Answer 1

We know, Area of circle $A=\pi r^2$
The rate of change of the area of a circle with respect to its radius is
$\frac{dA}{dr}=\frac{d\left(\pi r^2\right)}{dr}$

$=\pi \frac{d\left(r^2\right)}{dr}$

$=\pi \left(2r\right)$

$=2\pi r$

When $r=5\text{cm}$

$\frac{dA}{dr}=2\pi \left(5\right)=10\pi {\text{cm}}^2/\text{s}$

Hence, rate of change of area of the circle

when radius is $5\text{cm}$ is $10\pi {\text{cm}}^2/\text{s}$