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