Q 11. - Calculus

Question

If the volume and equator of a sphere are functions of time, what is the relationship between the rate of change of the volume of the sphere and the rate of change of the equator of the sphere? 

Step-by-Step Solution

Answer

The required relationship between the rate of change of the volume of the sphere and the rate of change of the equator of the sphere is $V^{\text{'}}\left(t\right)=2E^{\text{'}}\left(t\right){\left(r\left(t\right)\right)}^2$.

Step 1. Given information and formulas

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

And equator of sphere is $E=2\pi r$

Therefore, $V=V\left(t\right)$ and $E=E\left(t\right)$

Step 2. Explanation

Now $V\left(t\right)=\frac{4}{3}\pi {\left(r\left(t\right)\right)}^3$

Differentiate both sides with respect to $t$

$\Rightarrow V^{\text{'}}\left(t\right)=4\pi {\left(r\left(t\right)\right)}^2{r}^{\text{'}}\left(t\right) \cdots \cdots \left(i\right)$

Also $E\left(t\right)=2\pi r\left(t\right)$

Differentiate both sides with respect to $t$

$\Rightarrow E^{\text{'}}\left(t\right)=2\pi r^{\text{'}}\left(t\right) \cdots \cdots \left(ii\right)$

From $\left(i\right)$and $\left(ii\right)$

$\Rightarrow \frac{1}{4\pi {\left(r\left(t\right)\right)}^2}{V}^{\text{'}}\left(t\right)=\frac{1}{2\pi }{E}^{\text{'}}\left(t\right)$

$\Rightarrow V^{\text{'}}\left(t\right)=2E^{\text{'}}\left(t\right){\left(r\left(t\right)\right)}^2$