To find the diameter of a hemisphere with a given volume, we need to first find the radius and then double it to get the diameter. The formula to calculate the volume of a hemisphere is given by:
\[ \text{Volume of hemisphere} = \left(\frac{2}{3}\right) \pi r^3 \]
Given that the volume of the hemisphere is 458 cubic feet, we can set up the equation and solve for the radius \( r \):
\[ 458 = \left(\frac{2}{3}\right) \pi r^3 \]
To solve for \( r \), we need to isolate \( r \) on one side of the equation.
First, multiply both sides of the equation by the reciprocal of \( \left(\frac{2}{3}\right) \pi \) to get:
\[ r^3 = \left(\frac{458}{\left(\frac{2}{3}\right) \pi}\right) \]
\[ r^3 = \left(\frac{458 \cdot 3}{2 \pi}\right) \]
Now, we'll take the cube root of both sides to find \( r \):
\[ r = \left(\frac{458 \cdot 3}{2 \pi}\right)^{1/3} \]
Let's compute the right side of the equation step by step:
\[ r = \left(\frac{1374}{2 \pi}\right)^{1/3} \]
\[ r = \left(\frac{687}{\pi}\right)^{1/3} \]
For the purposes of calculation, we will use the approximation \( \pi \approx 3.14159 \).
\[ r = \left(\frac{687}{3.14159}\right)^{1/3} \]
\[ r \approx (218.60799)^{1/3} \]
\[ r \approx 6.024 \]
Finally, since the diameter is twice the radius, we have:
\[ \text{Diameter} = 2 \cdot r \]
\[ \text{Diameter} \approx 2 \cdot 6.024 \]
\[ \text{Diameter} \approx 12.048 \]
Rounding to the nearest tenth, the diameter of the hemisphere is approximately:
\[ \text{Diameter} \approx 12.0 \text{ ft} \]
So the diameter of the hemisphere, to the nearest tenth of a foot, is 12.0 feet.
