The volume of air inside a rubber ball with radius [tex]r[/tex] can be found using the function [tex]v(r) = \frac{4}{3} \pi r^3[/tex]. What does [tex]v\left(\frac{5}{7}\right)[/tex] represent?

A. The radius of the rubber ball when the volume equals [tex]\frac{5}{7}[/tex] cubic feet.
B. The volume of the rubber ball when the radius equals [tex]\frac{5}{7}[/tex] feet.
C. The volume of the rubber ball is 5 cubic feet when the radius is 7 feet.
D. The volume of the rubber ball is 7 cubic feet when the radius is 5 feet.



Answer :

To solve this problem, we first need to identify what the function \( v(r) = \frac{4}{3} \pi r^3 \) represents. This function is the formula for the volume of a sphere given its radius \( r \).

Now, we need to evaluate the function when the radius \( r \) is \( \frac{5}{7} \) feet. Substituting \( r = \frac{5}{7} \) into the formula:

[tex]\[ v\left(\frac{5}{7}\right) = \frac{4}{3} \pi \left( \frac{5}{7} \right)^3 \][/tex]

Let's write down the steps to compute this volume.

1. Calculate \( \left( \frac{5}{7} \right)^3 \):
[tex]\[ \left( \frac{5}{7} \right)^3 = \frac{5^3}{7^3} = \frac{125}{343} \][/tex]

2. Next, multiply this result by \( \frac{4}{3} \):
[tex]\[ \frac{4}{3} \times \frac{125}{343} = \frac{4 \times 125}{3 \times 343} = \frac{500}{1029} \][/tex]

3. Finally, multiply by \( \pi \):
[tex]\[ \frac{500}{1029} \pi \][/tex]

When this computation is performed, the numerical result is approximately 1.526527042560638.

Therefore, \( v\left(\frac{5}{7}\right) \) represents the volume of the rubber ball when the radius is \( \frac{5}{7} \) feet.

So, the correct answer is:
- The volume of the rubber ball when the radius equals [tex]\(\frac{5}{7}\)[/tex] feet