Question

The radius of a sphere is changing at the rate of 0.1 cm/sec. The rate of change of its surface area when the radius is 200 cm is
(a) 8π cm²/sec
(b) 12π cm²/sec
(c) 160π cm²/sec
(d) 200 cm²/sec

Answer 1

(c) 160π cm²/sec

$$\begin{gathered} \text{Let}\mathit{\text{r}}\text{be the radius and}\mathit{\text{S}}\text{be the surface area of the sphere at any time}\mathit{\text{t.}}\text{Then,} \\ S=\mathrm{4\pi }{r}^2 \\ \text{⇒}\frac{dS}{dt}\text{=8π}\mathit{\text{r}}\frac{dr}{dt} \\ \text{⇒}\frac{dS}{dt}\text{=8π}\left(200\right)\left(0.1\right) \\ \text{⇒}\frac{dS}{dt}{\text{=160π cm}}^2\text{/sec} \end{gathered}$$