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
Answer & explanation
Correct answer: option 4
Let r be the radius and V be the volume of the sphere. Then,
$V =\frac{4}{3} \pi r^3$
$\Rightarrow \frac{d V}{d t} =4 \pi r^2 \frac{d r}{d t}$
$\Rightarrow \frac{d r}{d t} =4 \pi r^2 \frac{d r}{d t}$ [∵ $\frac{d V}{d t}=\frac{d r}{d t}$ (given)]
$\Rightarrow 4 \pi r^2 =1 \Rightarrow r=\frac{1}{2 \sqrt{\pi}}$