Chapter 6, Problem 12STP

(a)

To determine

To Find :

The radius of the sphere.

Answer to Problem 12STP

12.7 inches

Explanation of Solution

Given :

The radius of a sphere with volume V can be found using the formula $r=\sqrt[3]{\frac{3V}{4\pi }}$ . V =8580 cu. in.

Calculation:

  $\begin{array}{l}Given\\ r=\sqrt[3]{\frac{3V}{4\pi }}\\ Substitute\text{ V=8580}\\ \text{⇒}r=\sqrt[3]{\frac{3(8580)}{4\pi }}=\sqrt[3]{\frac{25740}{4\pi }}=\sqrt[3]{2048.324118}=12.699\approx 12.7\\ \text{So, the radius of the sphere is 12}\text{.7 inches}\text{.}\end{array}$

(b)

To determine

To Find :

The formula for the volume of a sphere, given its radius .

Answer to Problem 12STP

  $V=\frac{4}{3}\pi r^3$

Explanation of Solution

Given :

The radius of a sphere with volume V can be found using the formula $r=\sqrt[3]{\frac{3V}{4\pi }}$ .

Calculation:

  $\begin{array}{l}Given\\ r=\sqrt[3]{\frac{3V}{4\pi }}\\ r^3={\left(\sqrt[3]{\frac{3V}{4\pi }}\right)}^3\mathrm{..............}[\text{eliminate the radical]}\\ r^3=\frac{3V}{4\pi }\\ r^3(4\pi )=3V\mathrm{...............}[\text{multiply each side by 4}\pi \text{]}\\ \text{4}\pi {\text{r}}^3=3V\\ \frac{3V}{3}=\frac{\text{4}\pi {\text{r}}^3}{3}\mathrm{.................}[\text{divide each side by 3]}\\ V=\frac{4}{3}\pi r^3\end{array}$

(c)

To determine

To Find :

The volume of the basketball that has diameter 9 inches.

Answer to Problem 12STP

  $V\approx 382i{n}^3$

Explanation of Solution

Given :

Diameter = 9 inches

From part (b) , $V=\frac{4}{3}\pi r^3$

Calculation:

  $\begin{array}{l}Given\\ V=\frac{4}{3}\pi r^3\\ \text{Substitute r = }\frac{diameter}{2}=\frac{9}{2}\\ V=\frac{4}{3}\pi {\left(\frac{9}{2}\right)}^3\\ V=\frac{4}{3}\pi \left(\frac{729}{8}\right)=\frac{243}{2}\pi =381.70\approx 382i{n}^3\end{array}$