Question

The radius of a sphere is increasing at the rate of 0.2 cm/sec. The rate at which the volume of the sphere increase when radius is 15 cm, is
(a) 12π cm³/sec
(b) 180π cm³/sec
(c) 225π cm³/sec
(d) 3π cm³/sec

Answer 1

(b) 180π cm³/sec

$$\begin{gathered} \text{Let}\mathit{\text{r}}\text{be the radius and}\mathit{\text{V}}\text{be the volume of the sphere at any time}\mathit{\text{t.}}\text{Then,} \\ \text{V=}\frac{4}{3}\text{π}{\mathit{\text{r}}}^3 \\ \text{⇒}\frac{dV}{dt}\text{=4π}{\mathit{\text{r}}}^2\frac{dr}{dt} \\ \text{⇒}\frac{dV}{dt}\text{=}\mathrm{4\pi }{\left(15\right)}^2\times 0.2 \\ \text{⇒}\frac{dV}{dt}{\text{=180π cm}}^3\text{/sec} \end{gathered}$$