From the statement, we know that the silo has two parts:
• the top is a hemisphere with a radius r = 5 ft,
,• the body is a right circular cylinder with a radius r = 5ft, and height h,
,• the total capacity of the silo is a volume Vₜ = 614 π ft³.
The total volume of the silo Vₜ is given by the sum of the volume of each part:
[tex]V_t=V_h+V_c=614\pi\times ft^3.[/tex]Where Vₕ is the volume of the hemisphere and Vc is the volume of the cylinder.
(1) The volume of the hemisphere is given by:
[tex]V_h=\frac{1}{2}\times V_s=\frac{1}{2}\times\frac{4}{3}\pi r^3=\frac{2}{3}\pi r^3=\frac{2}{3}\pi\times(5ft)^3=\frac{250}{3}\pi\times ft^3.[/tex](2) The volume of the cylinder is given by:
[tex]V_c=\pi r^2\times h=\pi\times(5ft)^2\times h=25\pi\times h\times ft^2.[/tex](3) Replacing the results from points (1) and (2) in the equation of the total volume, we have:
[tex]\frac{250}{3}\pi\times ft^3+25\pi\times h\times ft^2=614\pi\times ft^3.[/tex]Solving for h, we get:
[tex]\begin{gathered} \frac{250}{3}\pi ft^3+25\pi hft^2=614\pi ft^3, \\ 25\pi hft^2=614\pi ft^3-\frac{250}{3}\pi ft^3, \\ h=\frac{1}{25}*(614-\frac{250}{3})ft\cong21.23ft. \end{gathered}[/tex]AnswerThe height of the silo is 21.23 ft to the nearest hundredth of a ft.
