Question

a buoy is constructed out of the bottom half of a sphere with a cone on top. the radius of the sphere and the radius of the cone is 9 ft. the height of the buoy is 15 ft. what is the volume of the buoy? enter your answer, in exact form, in the box. ft³

Explanation:

Step1: Find the height of the cone

The height of the buoy is \(15\) ft and the radius of the hemisphere (which is the same as the radius of the sphere) is \(9\) ft. So the height of the cone \(h = 15 - 9=6\) ft.

Step2: Calculate the volume of the hemisphere

The formula for the volume of a sphere is \(V_{sphere}=\frac{4}{3}\pi r^{3}\). For a hemisphere \(V_{hemisphere}=\frac{2}{3}\pi r^{3}\). Given \(r = 9\) ft, then \(V_{hemisphere}=\frac{2}{3}\pi(9)^{3}=\frac{2}{3}\pi\times729 = 486\pi\) \(ft^{3}\).

Step3: Calculate the volume of the cone

The formula for the volume of a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h\). Here \(r = 9\) ft and \(h = 6\) ft. So \(V_{cone}=\frac{1}{3}\pi(9)^{2}\times6=\frac{1}{3}\pi\times81\times 6=162\pi\) \(ft^{3}\).

Step4: Calculate the volume of the buoy

The volume of the buoy \(V = V_{hemisphere}+V_{cone}\). Substitute the values: \(V=486\pi+162\pi=(486 + 162)\pi=648\pi\) \(ft^{3}\).

Answer:

\(648\pi\)