Question

The rate of change of volume of a sphere with respect to its surface area when the radius is $4$ cm is

• A. $6$ cm${}^3$/cm${}^2$
• B. $8$ cm${}^3$/cm${}^2$
• C. $2$ cm${}^3$/cm${}^2$
• D. $4$ cm${}^3$/cm${}^2$

Answer 1

The correct option is C $2$ cm${}^3$/cm${}^2$

$V=\frac{4}{3}\pi r^3$
$S=4\pi r^2$
$\frac{dV}{dS}=\frac{\frac{dV}{dr}}{\frac{dS}{dr}}$

$\frac{dV}{dr}==3\times \frac{4}{3}\pi r^2=4\pi r^2$
$\frac{dS}{dr}=8\pi r$
$\frac{dV}{dS}=\frac{4\pi r^2}{8\pi r}=\frac{r}{2}$
${\left[\frac{dV}{dS}\right]}_{r=4}=\frac{4}{2}=2$ cm${}^3$/cm${}^2$