Chapter 8.6, Problem 52HP

To determine

The volume of the air inside the ball. The formula is $v=\frac{4}{3}\pi r^3$ .

Answer to Problem 52HP

  $v=364$

Explanation of Solution

Given information:

Circumference of an NCAA women’s basketball is 29 inches and rubber coating is 3/16 inches thick.

  $v=\frac{3}{4}\pi r^3$ .

Calculation:

In order to find the volume of the air inside, first find the radius of the interior of the ball. The circumference of the ball is 29 inches, putting the value of C in the equation $C=2\pi r$and solve for r :

  $\begin{array}{l}C=2\pi r\\ 29=2\times 3.14\times r\\ 29=6.28r\end{array}$

now divide both sides by 6.28 for the value of r:

  $\begin{array}{l}29=6.28r\\ \frac{29}{6.28}=\frac{6.28r}{6.28}\\ r=4.61549\end{array}$

The radius of the ball is about 4.61549 inches. Now to find the radius of the interior of the ball, first subtract 3/16 from 4.61549, because the rubber coating is 3/16 inch thick, and then substitute the value of radius of the interior ball in the formula of the volume, that is $v=\frac{3}{4}\pi r^3$ and then solve for v:

  $\begin{array}{l}r=4.61549\\ \text{radius}\text{of}\text{interior of the ball}=4.61549-\frac{3}{16}\\ r=4.4225\\ v=\frac{4}{3}\pi r^3\\ v=\frac{4}{3}\times 3.14\times {4.4225}^3\\ v\approx 364\end{array}$

Thus, the volume of the air inside the ball is approximately 364 cubic inches.