Question

In a sphere the rate of change of surface area is
(a) 8π times the rate of change of diameter
(b) 2π times the rate of change of diameter
(c) 2π times the rate of change of radius
(d) 8π times the rate of change of radius

Answer 1

(d) 8π times the rate of change of radius

$$\begin{gathered} \text{Let}\mathit{\text{r}}\text{be the radius and}\mathit{\text{S}}\text{be the surface area of the sphere at any time}\mathit{\text{t.}}\text{Then,} \\ S=4\mathrm{\pi }{r}^2 \\ \Rightarrow \frac{dS}{dt}=8\mathrm{\pi }r\frac{dr}{dt} \\ \therefore \text{The rate of change of surface area is 8π times the rate of change of the radius.} \end{gathered}$$