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