Chapter P.3, Problem 72E

Solution

The formula for $r$ from the formula for the volume $V=\frac{4}{3}\pi r^3$ .

  $r=\sqrt[3]{\frac{3V}{4\pi }}$

Given:

Volume of a sphere $V=\frac{4}{3}\pi r^3$

Calculation:

In order to determine the formula for $r$ from the given volume of a sphere $V=\frac{4}{3}\pi r^3$ , first isolate this variable to one side of the equation such that $r$ can be solved as shown below:

  $V=\frac{4}{3}\pi r^3$

Multiply both sides by $\frac{3}{4}$ .

  $\begin{array}{l}V=\frac{4}{3}\pi r^3\\ V\times \frac{3}{4}=\frac{4}{3}\times \frac{3}{4}\pi r^3\\ \frac{3}{4}V=\pi r^3\end{array}$

Multiply both sides by $\pi$ .

  $\begin{array}{l}\frac{3}{4}V=\pi r^3\\ \frac{3V}{4\pi }=\frac{\pi }{\pi }{r}^3\\ \frac{3V}{4\pi }=r^3\end{array}$

Using cube root bot side

  $\begin{array}{l}\frac{3V}{4\pi }=r^3\\ \sqrt[3]{\frac{3V}{4\pi }}=\sqrt[3]{r^3}\\ r=\sqrt[3]{\frac{3V}{4\pi }}\end{array}$

So, the value of $r$ is $r=\sqrt[3]{\frac{3V}{4\pi }}$