Question

Find the rate of change of the volume of a sphere with respect to its surface area when the radius is 2 cm.

Answer 1

Let V be the volume of the sphere. Then,

V = $\frac{4}{3}\mathrm{\pi }{r}^3$

$\Rightarrow \frac{dV}{dr}=4\mathrm{\pi }{r}^2$

Let S be the total surface area of sphere. Then,

S = $4\mathrm{\pi }{r}^2$

$\Rightarrow \frac{dS}{dr}=8\mathrm{\pi }r$
$$\begin{gathered} \therefore \frac{dV}{dS}=\raisebox{1ex}{$\frac{dV}{dr}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle \frac{dS}{dr}}$}\right. \\ \Rightarrow \frac{dV}{dS}=\frac{4\mathrm{\pi }{r}^2}{8\mathrm{\pi }r}=\frac{r}{2} \\ \Rightarrow {\left(\frac{dV}{dS}\right)}_{r=2}=\frac{2}{2} \\ =1\mathrm{cm} \end{gathered}$$