Chapter SH, Problem 6.7EP

a.

To determine

Find the inverse of the given function.

Answer to Problem 6.7EP

The inverse of the given function is $v^{-1}=\sqrt[3]{\frac{3}{4\pi }r}$ .

Explanation of Solution

Given:

The given function is $v=\frac{4}{3}\pi r^3$ .

Calculation:

  $v=\frac{4}{3}\pi r^3$

For the inverse function, interchange the variables $v$ and $r$ .

  $r=\frac{4}{3}\pi v^3$

Multiply both sides by $3$ .

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

Divide both sides by $4\pi$ .

  $\begin{array}{l}\Rightarrow \frac{3}{4\pi }r=\frac{4\pi }{4\pi }{v}^3\\ \Rightarrow \frac{3}{4\pi }r=v^3\\ \Rightarrow v^3=\frac{3}{4\pi }r\\ \Rightarrow v={\left(\frac{3}{4\pi }r\right)}^{\frac{1}{3}}\\ \Rightarrow v=\sqrt[3]{\frac{3}{4\pi }r}\\ v^{-1}=\sqrt[3]{\frac{3}{4\pi }r}\end{array}$

Hence the inverse of the given function is $v^{-1}=\sqrt[3]{\frac{3}{4\pi }r}$ .

b.

To determine

Explain the meaning of the inverse function in the given situation.

Explanation of Solution

Given:

The given function is $v=\frac{4}{3}\pi r^3$ .

Calculation:

  $v=\frac{4}{3}\pi r^3$

For the inverse function, interchange the variables $v$ and $r$ .

  $r=\frac{4}{3}\pi v^3$

Multiply both sides by $3$ .

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

Divide both sides by $4\pi$ .

  $\begin{array}{l}\Rightarrow \frac{3}{4\pi }r=\frac{4\pi }{4\pi }{v}^3\\ \Rightarrow \frac{3}{4\pi }r=v^3\\ \Rightarrow v^3=\frac{3}{4\pi }r\\ \Rightarrow v={\left(\frac{3}{4\pi }r\right)}^{\frac{1}{3}}\\ \Rightarrow v=\sqrt[3]{\frac{3}{4\pi }r}\\ v^{-1}=\sqrt[3]{\frac{3}{4\pi }r}\end{array}$

In the original function, $v$ represents the volume and $r$ represents the radius of the sphere but in the inverse function $r$ represents the volume and $v$ represents the radius of the sphere