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

Answer 1

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

$$\begin{gathered} \text{Let}\mathit{\text{r}}\text{be the radius and}\mathit{\text{V}}\text{be the volume of sphere at any time}\mathit{\text{t.}}\text{Then,} \\ V=\frac{4}{3}\text{π}{r}^3 \\ \Rightarrow \frac{dV}{dt}=\frac{4}{3}\left(3\text{π}{\mathit{\text{r}}}^2\right)\left(\frac{dr}{dt}\right) \\ \Rightarrow \frac{dV}{dt}=4\mathrm{\pi }{r}^2\left(\frac{dr}{dt}\right) \\ \mathrm{Thus},\mathrm{t}\text{he rate of change of volume is surface area times the rate of change of the radius.} \end{gathered}$$