Question

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

• A. $ 4 \, cm^3 / cm^2$
• B. $ 2 \, cm^3 / cm^2$
• C. $6 \, cm^3 / cm^2$
• D. $8 \, cm^3 / cm^2$

Answer 1

Answer: (B)

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