Question

A balloon that always remains spherical has a variable radius.

The rate at which its volume is increasing is $m\pi c{m}^2$ with the radius when the latter is $10cm$.

Find $m$.

Answer 1

Solve for the required value:

Given: The rate at which the volume of the sphere is increasing is $m\pi c{m}^2$, and the radius is $10cm$

The volume of the sphere is $V=\frac{4}{3}{\mathrm{\pi r}}^3$

The rate of change of the volume with respect to radius is $\frac{dV}{dr}$

$$\begin{gathered} \frac{\mathrm{dV}}{\mathrm{dr}}=\frac{d}{dr}\left(\frac{4}{3}{\mathrm{\pi r}}^3\right) \\ \Rightarrow \frac{\mathrm{dV}}{\mathrm{dr}}=\frac{4}{3}\mathrm{\pi }\times 3{\mathrm{r}}^2 \\ \Rightarrow \frac{\mathrm{dV}}{\mathrm{dr}}=4{\mathrm{\pi r}}^2 \\ \Rightarrow \frac{\mathrm{dV}}{\mathrm{dr}}=4\mathrm{\pi }{\left(10\right)}^2\left[\because \mathrm{r}=10\right] \\ \Rightarrow \frac{\mathrm{dV}}{\mathrm{dr}}=400\mathrm{\pi } \end{gathered}$$

Hence, the value of $m$ is $400$