To solve for the volume of the sphere when a cube is inscribed in it, and given that the cube's volume is 125 cubic units, let's follow these steps:
1. Calculate the side length of the cube:
The volume [tex]\( V \)[/tex] of a cube is given by:
[tex]\[
V = s^3
\][/tex]
where [tex]\( s \)[/tex] is the side length. Given [tex]\( V = 125 \)[/tex]:
[tex]\[
s^3 = 125 \implies s = \sqrt[3]{125} = 5 \text{ units}
\][/tex]
2. Calculate the diagonal of the cube:
The diagonal [tex]\( d \)[/tex] of a cube relates to its side length by the formula:
[tex]\[
d = s\sqrt{3}
\][/tex]
Substituting [tex]\( s = 5 \)[/tex]:
[tex]\[
d = 5\sqrt{3}
\][/tex]
3. Determine the radius of the sphere:
Since the sphere is circumscribed around the cube, its diameter is equal to the diagonal of the cube. Thus, the radius [tex]\( r \)[/tex] of the sphere is:
[tex]\[
r = \frac{d}{2} = \frac{5\sqrt{3}}{2}
\][/tex]
4. Calculate the volume of the sphere:
The volume [tex]\( V \)[/tex] of a sphere is given by:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
Substituting [tex]\( r = \frac{5\sqrt{3}}{2} \)[/tex]:
[tex]\[
V = \frac{4}{3} \pi \left( \frac{5\sqrt{3}}{2} \right)^3
\][/tex]
Simplifying the term inside the cube:
[tex]\[
\left( \frac{5\sqrt{3}}{2} \right)^3 = \frac{(5\sqrt{3})^3}{2^3} = \frac{125 \cdot 3\sqrt{3}}{8} = \frac{375\sqrt{3}}{8}
\][/tex]
Now substituting back into the volume formula:
[tex]\[
V = \frac{4}{3} \pi \left( \frac{375\sqrt{3}}{8} \right)
\][/tex]
Simplifying further:
[tex]\[
V = \frac{4 \cdot 375\sqrt{3} \pi}{24} = \frac{125 \pi \sqrt{3}}{2}
\][/tex]
Thus, the volume of the sphere is:
[tex]\[
\boxed{\frac{125 \pi \sqrt{3}}{2}}
\][/tex]
