In a sphere the rate of change of volume is

Question:

In a sphere the rate of change of volume is
(a) π times the rate of change of radius
(b) surface area times the rate of change of diameter
(c) surface area times the rate of change of radius
(d) none of these

Solution:

(c) surface area times the rate of change of radius

Let r be the radius and V be the volume of sphere at any time t. Then,

$V=\frac{4}{3} \pi r^{3}$

$\Rightarrow \frac{d V}{d t}=\frac{4}{3}\left(3 \pi r^{2}\right)\left(\frac{d r}{d t}\right)$

$\Rightarrow \frac{d V}{d t}=4 \pi r^{2}\left(\frac{d r}{d t}\right)$

Thus, the rate of change of volume is surface area times the rate of change of the radius.

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