Q. 13 - Calculus

Question

Suppose the radius $r$, volume $V$, and surface area $S$ of a sphere are functions of time $t$.

(a) How are $\frac{dV}{dt}$ and $\frac{dr}{dt}$  related?

(b) How are $\frac{dS}{dt}$ and $\frac{dr}{dt}$ related? 

Step-by-Step Solution

Answer

(a)The derivative $\frac{dV}{dt}$ and $\frac{dr}{dt}$ are related by$\frac{dV}{dt}=4{\mathrm{\pi r}}^2\frac{\mathrm{dr}}{\mathrm{dt}}$.

(b)The derivative $\frac{dS}{dt}$ and $\frac{dr}{dt}$ are related by $\frac{dS}{dt}=8\mathrm{\pi r}\frac{\mathrm{dr}}{\mathrm{dt}}$.

Part (a) Step 1. Formula used.

The volume V of the sphere is given by the formula

$V=\frac{4}{3}{\mathrm{\pi r}}^3$ cu. units.

Part (a) Step 2. Apply the differentiation.

Since V  and r  are functions of t.

Apply the differentiation to $V=\frac{4}{3}{\mathrm{\pi r}}^3$ with respect to t as follows.

$$\begin{gathered} \frac{dV}{dt}=\frac{4}{3}\mathrm{\pi }\left(3{\mathrm{r}}^2\frac{\mathrm{dr}}{\mathrm{dt}}\right) \\ \frac{\mathrm{dV}}{\mathrm{dt}}=4{\mathrm{\pi r}}^2\frac{\mathrm{dr}}{\mathrm{dt}} \end{gathered}$$

Part (a) Step 3. Conclusion

The derivatives $\frac{dV}{dt}$ and $\frac{dr}{dt}$ are related by $\frac{\mathrm{dV}}{\mathrm{dt}}=4{\mathrm{\pi r}}^2\frac{\mathrm{dr}}{\mathrm{dt}}$.

Part (b) Step 1. Formula used.

The surface area of the sphere is given by the formula $S=4{\mathrm{\pi r}}^2$ sq. units.

Part (b) Step 2. Apply the differentiation.

Given S and r are functions of t.

Apply the differentiation to $S=4{\mathrm{\pi r}}^2$ with respect to t as follows.

$$\begin{gathered} \frac{dS}{dt}=4\mathrm{\pi }\left(2\mathrm{r}\frac{\mathrm{dr}}{\mathrm{dt}}\right) \\ \frac{\mathrm{dS}}{\mathrm{dt}}=8\mathrm{\pi r}\frac{\mathrm{dr}}{\mathrm{dt}} \end{gathered}$$

Part (b) Step 3. Conclusion.

The derivatives $\frac{dS}{dt}$ and $\frac{dr}{dt}$ are related by $\frac{\mathrm{dS}}{\mathrm{dt}}=8\mathrm{\pi r}\frac{\mathrm{dr}}{\mathrm{dt}}$.