Question

The rate of change of volume of sphere with respect to its surface area $S$ is

• A. $\sqrt{\frac{S}{\pi }}$
• B. $\frac{1}{2}\sqrt{\frac{S}{\pi }}$
• C. $\frac{1}{4}\sqrt{\frac{S}{\pi }}$
• D. $4\sqrt{\frac{S}{\pi }}$

Answer 1

The correct option is C $\frac{1}{4}\sqrt{\frac{S}{\pi }}$

Volume of sphere with radius $r$ is $\frac{4}{3}\pi r^3$ and surface area $S=\pi r^2$
$\therefore \frac{dV}{dr}=\frac{4}{3}\pi \left(3r^2\right)=4\pi r^2$
Now $S=4\pi r^2$
$\frac{dS}{dr}=4\pi \left(2r\right)$
$\therefore \frac{dV}{dS}=\frac{\frac{dV}{dS}}{\frac{dS}{dr}}=\frac{4\pi r^2}{8\pi r}=\frac{r}{2}$
$=\frac{1}{2}\left(r\right)...\left(1\right)$
Now $S=4\pi r^2$
$\therefore r^2=\frac{S}{4\pi }$
$r=\frac{1}{2}\left(\sqrt{\frac{S}{\pi }}\right)$
$\therefore$ Rate of change of volume w,r,to
$S=\frac{1}{2}\left[\frac{1}{2}\left(\sqrt{\frac{S}{\pi }}\right)\right]=\frac{1}{4}\left(\sqrt{\frac{S}{\pi }}\right)$