Question

a water tank, in the shape of a cone on top of a cylinder, is shown below. the height of the water tank is 18 meters. the height of the cylindrical portion is 9 meters, and the diameter of the base is 12 meters. what is the volume of water that this tank can hold? leave your answer in terms of \\( \pi \\). 1 of 4 questions \\( v = 648\pi m^{3} \\) \\( v = 288\pi m^{3} \\) \\( v = 486\pi m^{3} \\) \\( v = 432\pi m^{3} \\)

Explanation:

Step1: Calculate the volume of the cylinder

The formula for the volume of a cylinder is \(V_{cylinder}=\pi r^{2}h\). Given \(r = 6\) (since diameter \(d = 12\), \(r=\frac{d}{2}\)) and \(h = 9\).
\(V_{cylinder}=\pi\times6^{2}\times9=\pi\times36\times9 = 324\pi\)

Step2: Calculate the height of the cone

The total height of the tank is \(18\) meters and the height of the cylinder is \(9\) meters. So the height of the cone \(h_{cone}=18 - 9=9\) meters.

Step3: Calculate the volume of the cone

The formula for the volume of a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h\). With \(r = 6\) and \(h = 9\)
\(V_{cone}=\frac{1}{3}\pi\times6^{2}\times9=\frac{1}{3}\pi\times36\times9= 108\pi\)

Step4: Calculate the total volume of the tank

\(V = V_{cylinder}+V_{cone}\)
\(V=324\pi + 108\pi=432\pi\)

Answer:

\(V = 432\pi m^{3}\) (the fourth option)