Question

Find the rate of change of the area of a circle with respect to its radius $r$ when
(a) $r=3\text{cm}$ (b) $r=4\text{cm}$

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 $r=3\text{cm}$

$\frac{dA}{dr}=2\pi \left(3\right)=6\pi {\text{cm}}^2/\text{s}$

When $r=4\text{cm}$

$\frac{dA}{dr}=2\pi \left(4\right)=8\pi {\text{cm}}^2/\text{s}$

Hence, rate of change of area of the circle when radius is $3\text{cm}$ is $6\pi {\text{cm}}^2/\text{s}$ and when radius is $4\text{cm}$ is $8\pi {\text{cm}}^2/\text{s}$