To solve for [tex]\(\frac{V}{\pi}\)[/tex] given the surface area of the sphere, we start by using the formula for the surface area of a sphere:
[tex]\[ A = 4 \pi r^2 \][/tex]
Here, the surface area [tex]\( A \)[/tex] is given as [tex]\( 2500 \pi \)[/tex] square inches. Substituting [tex]\( A = 2500 \pi \)[/tex] into the formula, we have:
[tex]\[ 2500 \pi = 4 \pi r^2 \][/tex]
Next, we need to solve for the radius [tex]\( r \)[/tex]. First, divide both sides by [tex]\( 4 \pi \)[/tex]:
[tex]\[ r^2 = \frac{2500 \pi}{4 \pi} \][/tex]
Simplify the right-hand side:
[tex]\[ r^2 = \frac{2500}{4} = 625 \][/tex]
Now solve for [tex]\( r \)[/tex] by taking the square root of both sides:
[tex]\[ r = \sqrt{625} \][/tex]
[tex]\[ r = 25 \][/tex]
With the radius [tex]\( r \)[/tex] now known, we can find the volume [tex]\( V \)[/tex] of the sphere using the formula:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Substitute [tex]\( r = 25 \)[/tex] into this formula:
[tex]\[ V = \frac{4}{3} \pi (25)^3 \][/tex]
Now calculate [tex]\( (25)^3 \)[/tex]:
[tex]\[ 25^3 = 25 \times 25 \times 25 \][/tex]
[tex]\[ 25 \times 25 = 625 \][/tex]
[tex]\[ 625 \times 25 = 15625 \][/tex]
So:
[tex]\[ V = \frac{4}{3} \pi \times 15625 \][/tex]
Now simplify the expression inside the volume formula:
[tex]\[ V = \frac{4 \times 15625}{3} \pi \][/tex]
[tex]\[ V = \frac{62500}{3} \pi \][/tex]
Next, we need [tex]\(\frac{V}{\pi}\)[/tex]:
[tex]\[ \frac{V}{\pi} = \frac{\frac{62500}{3} \pi}{\pi} \][/tex]
The [tex]\(\pi\)[/tex] terms cancel out:
[tex]\[ \frac{V}{\pi} = \frac{62500}{3} \][/tex]
Now compute [tex]\(\frac{62500}{3}\)[/tex]:
[tex]\[ \frac{62500}{3} \approx 20833.33 \][/tex]
Therefore, the value of [tex]\(\frac{V}{\pi}\)[/tex] is approximately:
[tex]\[ 20833.33 \][/tex]
So, the correct answer is:
[tex]\[ 20833.33 \ln^3 \][/tex]
