To find the radius of the cone given its volume and height, we can use the formula for the volume of a cone:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
Given:
- Volume [tex]\(V = 50 \pi\)[/tex] cubic feet
- Height [tex]\(h = 6\)[/tex] feet
We need to find the radius [tex]\(r\)[/tex]. First, solve the volume formula for [tex]\(r^2\)[/tex]:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
Rearrange to solve for [tex]\(r^2\)[/tex]:
[tex]\[
r^2 = \frac{3V}{\pi h}
\][/tex]
Substitute the given values into the equation:
[tex]\[
r^2 = \frac{3 \cdot 50 \pi}{\pi \cdot 6}
\][/tex]
Simplify the right-hand side:
[tex]\[
r^2 = \frac{150 \pi}{6 \pi} = \frac{150}{6} = 25
\][/tex]
Thus,
[tex]\[
r^2 = 25
\][/tex]
Now, take the square root of both sides to find [tex]\(r\)[/tex]:
[tex]\[
r = \sqrt{25} = 5
\][/tex]
Finally, we round the radius to the nearest tenth, but in this case, it is an exact number:
[tex]\[
\boxed{5.0}
\][/tex]
So, the radius of the cone is 5.0 feet.
