To find the radius of the sphere given a volume of [tex]\(\frac{500}{3} \pi \, \text{cm}^3\)[/tex], we follow these steps:
1. Substitute the given value into the volume formula:
The volume [tex]\( V \)[/tex] of a sphere is given by:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
We know the volume is [tex]\(\frac{500}{3} \pi \, \text{cm}^3\)[/tex], so we substitute this into the formula:
[tex]\[
\frac{500}{3} \pi = \frac{4}{3} \pi r^3
\][/tex]
2. Simplify the equation by dividing both sides by [tex]\(\frac{4}{3}\)[/tex]:
To isolate [tex]\(r^3\)[/tex], we need to undo the multiplication by [tex]\(\frac{4}{3}\)[/tex]:
[tex]\[
r^3 = \frac{\frac{500}{3} \pi}{\frac{4}{3} \pi}
\][/tex]
Since the [tex]\(\pi\)[/tex] terms cancel out, we have:
[tex]\[
r^3 = \frac{\frac{500}{3}}{\frac{4}{3}}
\][/tex]
3. Simplify the division:
Dividing by [tex]\(\frac{4}{3}\)[/tex] is the same as multiplying by [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
r^3 = \frac{500}{3} \times \frac{3}{4} = \frac{500 \times 3}{3 \times 4} = \frac{500}{4}
\][/tex]
So:
[tex]\[
r^3 = 125
\][/tex]
4. Find the cube root to solve for [tex]\(r\)[/tex]:
To find [tex]\(r\)[/tex], we take the cube root of both sides of the equation:
[tex]\[
r = \sqrt[3]{125}
\][/tex]
The cube root of 125 is 5, so:
The radius of the sphere is [tex]\( \boxed{5} \)[/tex] cm.
