The volume of a sphere is [tex]\frac{500}{3} \pi \, \text{cm}^3[/tex]. What is the radius?

The formula for the volume of a sphere is [tex]V = \frac{4}{3} \pi r^3[/tex].

1. Substitute the given value into the formula:
[tex]\[
\frac{500}{3} \pi = \frac{4}{3} \pi r^3
\][/tex]

2. Undo the multiplication of [tex]\frac{4}{3}[/tex]:
[tex]\[
125 \pi = \pi r^3
\][/tex]

3. Undo the multiplication of [tex]\pi[/tex]:
[tex]\[
125 = r^3
\][/tex]

4. Undo the cube:
[tex]\[
\sqrt[3]{125} = r
\][/tex]

The radius of the sphere is [tex] \boxed{5} \, \text{cm}[/tex].



Answer :

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.