To determine the volume of air that a soccer ball with a diameter of 9 inches can hold, we use the formula for the volume of a sphere:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
where [tex]\( V \)[/tex] is the volume and [tex]\( r \)[/tex] is the radius of the sphere.
Step-by-Step Solution:
1. Determine the radius:
The diameter of the soccer ball is given as 9 inches. The radius is half of the diameter.
[tex]\[
r = \frac{\text{diameter}}{2} = \frac{9 \text{ inches}}{2} = 4.5 \text{ inches}
\][/tex]
2. Substitute the radius into the volume formula:
[tex]\[
V = \frac{4}{3} \pi (r)^3
\][/tex]
We have already calculated the radius [tex]\( r = 4.5 \text{ inches} \)[/tex]. Now, substitute this value into the formula:
[tex]\[
V = \frac{4}{3} \pi (4.5)^3
\][/tex]
3. Calculate [tex]\( r^3 \)[/tex]:
[tex]\[
(4.5)^3 = 4.5 \times 4.5 \times 4.5 = 91.125
\][/tex]
4. Calculate the volume:
[tex]\[
V = \frac{4}{3} \pi \times 91.125
\][/tex]
[tex]\[
V = \frac{4}{3} \times 3.141592653589793 \times 91.125
\][/tex]
[tex]\[
V \approx 381.7035074111598 \text{ cubic inches}
\][/tex]
5. Round the volume to the nearest hundredth:
[tex]\[
V \approx 381.70 \text{ cubic inches}
\][/tex]
Thus, the volume of air that the soccer ball can hold, rounded to the nearest hundredth, is approximately 381.70 cubic inches.
