Answer:
Approximately \(71.85\) cubic inches.
Step-by-step explanation:
The volume of a sphere of radius \(r\) is \((4/3)\, \pi\, r^{3}\). The volume of a hemisphere would be half that amount: \((2/3)\, \pi\, r^{3}\).
The circumference of a sphere (or a hemisphere) of radius \(r\) at its equator is \(2\, \pi\, r\).
To find the volume of a hemisphere in terms of the circumference at the equator, apply the following steps:
- Express the radius of the hemisphere in terms of circumference at the equator.
- Substitute the expression for radius (in terms of the circumference) into the expression for volume.
Let \(C\) denote the circumference of this hemisphere at the equator. Given that \(C = 2\, \pi\, r\), rearrange this equation and solve for the radius \(r\) of the hemisphere:
\(\begin{aligned}r &= \frac{C}{2\, \pi}\end{aligned}\).
Substitute the expression for radius (in terms of circumference \(C\)) into the expression for the volume of the hemisphere:
\(\begin{aligned}V &= \frac{2}{3}\, \pi\, r^{3} \\ &= \frac{2}{3}\, (\pi)\, \left(\frac{C}{2\, \pi}\right)^{3} \\ &= \frac{C^{3}}{12\, \pi^{2}}\end{aligned}\).
Given that the circumference of this hemisphere at the equator is \(C = 92.25\) inches, the volume \(V\) (in cubic inches) of the hemisphere would be:
\(\begin{aligned}V &= \frac{C^{3}}{12\, \pi^{2}} = \frac{(92.25)^{3}}{12\, \pi^{2}} \approx 71.85 \end{aligned}\).
