Question:
If the rate of change of volume of a sphere is equal to the rate of change of its radius, then its radius is equal to
(a) 1 unit
(b) $\sqrt{2 \pi}$ units
(c) $\frac{1}{\sqrt{2 \pi}}$ unit
(d) $\frac{1}{2 \sqrt{\pi}}$ unit
Solution:
(d) $\frac{1}{2 \sqrt{\pi}}$ unit
Let $r$ be the radius and $V$ be the volume of the sphere at any time $t .$ Then,
$V=\frac{4}{3} \pi r^{3}$
$\Rightarrow \frac{d V}{d t}=\frac{4}{3}\left(3 \pi r^{2}\right) \frac{d r}{d t}$
$\Rightarrow \frac{d V}{d t}=4 \pi r^{2} \frac{d r}{d t}$
$\Rightarrow 4 \pi r^{2}=1 \quad\left[\because \frac{d V}{d t}=\frac{d r}{d t}\right]$
$\Rightarrow r^{2}=\frac{1}{4 \pi}$
$\Rightarrow r=\sqrt{\frac{1}{4 \pi}}$
$\Rightarrow r=\frac{1}{2 \sqrt{\pi}}$ unit