To determine the volume of a sphere, we use the formula:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
Given that the diameter of the sphere is [tex]\(5\)[/tex] meters, we can find the radius by dividing the diameter by [tex]\(2\)[/tex]:
[tex]\[
r = \frac{5}{2} = 2.5 \, \text{meters}
\][/tex]
Now, we substitute the radius back into the volume formula:
[tex]\[
V = \frac{4}{3} \pi (2.5)^3
\][/tex]
Calculating [tex]\(2.5\)[/tex] cubed:
[tex]\[
(2.5)^3 = 2.5 \times 2.5 \times 2.5 = 6.25 \times 2.5 = 15.625
\][/tex]
Thus, the volume formula becomes:
[tex]\[
V = \frac{4}{3} \pi \times 15.625
\][/tex]
Simplifying the calculation:
[tex]\[
V = \frac{4 \times 15.625}{3} \pi = \frac{62.5}{3} \pi
\][/tex]
Thus, the volume of the sphere is:
[tex]\[
V = \frac{62.5}{3} \pi \, \text{m}^3
\][/tex]
Next, let's convert [tex]\(\frac{62.5}{3}\)[/tex] to a fraction in the simplest form to match with the given multiple choices:
The term [tex]\(62.5\)[/tex] as a fraction equals [tex]\(\frac{625}{10}\)[/tex]. So:
[tex]\[
V = \frac{625}{10} \times \frac{\pi}{3} = \frac{625}{30} \pi = \frac{125}{6} \pi
\][/tex]
Thus, the correct answer matches with:
[tex]\[
V = \frac{125}{6} \pi \text{ m}^3
\][/tex]
Therefore, the volume of the sphere is:
[tex]\[
\boxed{\frac{125}{6} \pi \text{ m}^3}
\][/tex]
