To determine the diameter of a hemisphere given its volume, we can follow these steps:
1. Understand the relationship between volume and radius for a hemisphere:
The formula for the volume of a hemisphere is:
[tex]\[
V = \frac{2}{3} \pi r^3
\][/tex]
where [tex]\( V \)[/tex] is the volume and [tex]\( r \)[/tex] is the radius of the hemisphere.
2. Solving for the radius [tex]\( r \)[/tex]:
Rearrange the formula to solve for the radius:
[tex]\[
r^3 = \frac{3}{2} \frac{V}{\pi}
\][/tex]
Taking the cube root of both sides gives:
[tex]\[
r = \left( \frac{3}{2} \frac{V}{\pi} \right)^{\frac{1}{3}}
\][/tex]
3. Insert the given volume:
Plug in the given volume [tex]\( V = 8514 \text{ cm}^3 \)[/tex]:
[tex]\[
r = \left( \frac{3}{2} \frac{8514}{\pi} \right)^{\frac{1}{3}}
\][/tex]
4. Calculate the radius:
After performing the calculation, we get:
[tex]\[
r \approx 15.96 \text{ cm}
\][/tex]
5. Determine the diameter of the sphere:
Since the diameter [tex]\( D \)[/tex] of a sphere is twice the radius, we calculate it as:
[tex]\[
D = 2r \approx 2 \times 15.96 = 31.92 \text{ cm}
\][/tex]
6. Round the diameter to the nearest tenth:
Rounding 31.92 to the nearest tenth, we get:
[tex]\[
D \approx 31.9 \text{ cm}
\][/tex]
Thus, the diameter of the hemisphere, to the nearest tenth of a centimeter, is approximately 31.9 cm.
