Question

If the rate of change of volume of a sphere is equal to the rate of change of its radius, then its radius is equal to

(a) 1 unit

(b) $\sqrt{2\mathrm{\pi }}\mathrm{units}$

(c) $\frac{1}{\sqrt{2\mathrm{\pi }}}\mathrm{unit}$

(d) $\frac{1}{2\sqrt{\mathrm{\pi }}}\mathrm{unit}$

Answer 1

(d) $\frac{1}{2\sqrt{\mathrm{\pi }}}\mathrm{unit}$

$$\begin{gathered} \text{Let}\mathit{\text{r}}\text{be the radius and}\mathit{\text{V}}\text{be the volume of the sphere at any time}\mathit{\text{t}}\text{. Then,} \\ \mathit{\text{V}}\text{=}\frac{4}{3}\text{π}{\mathit{\text{r}}}^3 \\ \text{⇒}\frac{dV}{dt}\text{=}\frac{4}{3}\left(3\mathrm{\pi }{r}^2\right)\frac{dr}{dt} \\ \text{⇒}\frac{dV}{dt}\text{=4}\mathrm{\pi }{r}^2\frac{dr}{dt} \\ \Rightarrow 4\mathrm{\pi }{r}^2=1\left[\because \frac{dV}{dt}\text{=}\frac{dr}{dt}\right] \\ \Rightarrow r^2=\frac{1}{4\mathrm{\pi }} \\ \Rightarrow r=\sqrt{\frac{1}{4\mathrm{\pi }}} \\ \Rightarrow r=\frac{1}{2\sqrt{\mathrm{\pi }}}\mathrm{unit} \end{gathered}$$