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
