To find the radius of a sphere given its volume, we need to use the formula for the volume of a sphere:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
In this formula, [tex]\( V \)[/tex] represents the volume, and [tex]\( r \)[/tex] represents the radius.
Given:
[tex]\[ V = 36 \, \text{in}^3 \][/tex]
We need to solve for [tex]\( r \)[/tex]:
1. Start with the volume formula:
[tex]\[ 36 = \frac{4}{3} \pi r^3 \][/tex]
2. Rearrange the formula to solve for [tex]\( r^3 \)[/tex]:
[tex]\[ r^3 = \frac{36}{\frac{4}{3} \pi} \][/tex]
3. Simplify:
[tex]\[ r^3 = \frac{36 \times 3}{4 \pi} \][/tex]
[tex]\[ r^3 = \frac{108}{4 \pi} \][/tex]
[tex]\[ r^3 = \frac{27}{\pi} \][/tex]
4. To find [tex]\( r \)[/tex], we take the cube root of both sides:
[tex]\[ r = \sqrt[3]{\frac{27}{\pi}} \][/tex]
5. Calculate the value:
[tex]\[ r \approx 2.048352189765887 \][/tex]
6. Round the result to the nearest whole number:
[tex]\[ r \approx 2 \][/tex]
Therefore, the radius of the sphere, rounded to the nearest whole number, is [tex]\( 2 \)[/tex] inches.
