Question

If the rate of change in the volume of a sphere is equal to the rate of change in its radius, then the radius is equal to

• A. $\frac{1}{2\sqrt{\pi }}$
• B. $\frac{1}{2\pi }$
• C. $\frac{2}{\pi }$
• D. $\frac{1}{\sqrt{2\pi }}$

Answer 1

The correct option is A $\frac{1}{2\sqrt{\pi }}$

$\text{Volume of sphere},V=\frac{4}{3}\pi r^3$
$\text{Given},\frac{dV}{dt}=\frac{dr}{dt}$
$\Rightarrow \frac{4\pi }{3}\times 3r^2\times \frac{dr}{dt}=\frac{dr}{dt}$
$\Rightarrow r^2=\frac{1}{4\pi }\Rightarrow r=\frac{1}{2\sqrt{\pi }}$