Question

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

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

Answer 1

Answer: (B)

Let the radius of the sphere be a.
$\therefore$ Volume, $V=\frac{4}{3}\pi a^3$
$\Rightarrow \frac{dV}{da}=\frac{4}{3}\pi \left(3a^2\right)=4\pi a^2$
Again, surface area, $s=4\pi a^2$
$\Rightarrow \frac{ds}{da}=4\pi (2a)=8\pi a$
$\therefore \frac{dV}{ds}=\frac{(dV/da)}{(ds/da)}$
$=\frac{4\pi a^2}{8\pi a}=\frac{a}{2}$
$=\frac{4}{2}$
$[\because a=4cm]$
$=2c{m}^3/c{m}^2$