Question

The volume of metal in a hollow sphere is constant. If the inner radius is increasing at the rate of 1 cm/sec, find the rate of increase of the outer radius when the radii are 4 cm and 8 cm respectively.

Answer 1

$$\begin{gathered} \text{Let}{\mathit{\text{r}}}_1\text{be the inner radius and}{\mathit{\text{r}}}_2\text{be the outer radius and}\mathit{\text{V}}\text{be the volume}\mathrm{of}\mathrm{the}\mathrm{hollow}\mathrm{sphere}\text{at any time}\mathit{\text{t}}\text{. Then,} \\ \mathit{\text{V}}\text{=}\frac{4}{3}\text{π}\left({r_1}^3-{r_2}^3\right) \\ \Rightarrow \frac{dV}{dt}\text{=4π}\left({r_1}^2\frac{d{r}_1}{dt}-{r_2}^2\frac{d{r}_2}{dt}\right) \\ \text{⇒}{r_1}^2\frac{d{r}_1}{dt}={r_2}^2\frac{d{r}_2}{dt}\left[\because \frac{dV}{dt}=0\right] \\ \text{⇒}{\left(4\right)}^2\text{×1=}{\left(8\right)}^2\frac{d{r}_2}{dt} \\ \text{⇒}\frac{d{r}_2}{dt}\text{=}\frac{1}{4}\text{cm/sec} \end{gathered}$$