Question

a cylinder has radius 9 inches and height 15 inches. a cone has the same radius and height.
question 2 part a
a) find the volume of the cylinder in terms of pi cubic inches
question 2 part b
b) find the volume of the cone in terms of pi cubic inches
question 2 part c
what fraction of the cylinders volume is the cones volume?

Explanation:

Step1: Recall the volume formula for a cylinder

The volume formula for a cylinder is \(V = \pi r^{2}h\). Given \(r = 9\) inches and \(h=15\) inches.
Substitute the values into the formula: \(V=\pi\times9^{2}\times15\).
First, calculate \(9^{2}=81\). Then \(V = \pi\times81\times15\).
\(81\times15=(80 + 1)\times15=80\times15+1\times15=1200 + 15=1215\). So the volume of the cylinder \(V_{cylinder}=1215\pi\) cubic inches.

Step2: Recall the volume formula for a cone

The volume formula for a cone is \(V=\frac{1}{3}\pi r^{2}h\). Since \(r = 9\) inches and \(h = 15\) inches.
Substitute the values into the formula: \(V=\frac{1}{3}\pi\times9^{2}\times15\).
We know \(9^{2}=81\), then \(V=\frac{1}{3}\pi\times81\times15\).
\(\frac{1}{3}\times81 = 27\), and \(27\times15=(30 - 3)\times15=30\times15-3\times15=450-45 = 405\). So the volume of the cone \(V_{cone}=405\pi\) cubic inches.

Step3: Find the fraction of the cone's volume to the cylinder's volume

The fraction \(f=\frac{V_{cone}}{V_{cylinder}}\). Substitute \(V_{cone}=405\pi\) and \(V_{cylinder}=1215\pi\).
\(f=\frac{405\pi}{1215\pi}\). Cancel out \(\pi\) (since \(\pi
eq0\)), and \(\frac{405}{1215}=\frac{405\div405}{1215\div405}=\frac{1}{3}\).

Answer:

a) \(1215\)
b) \(405\)
c) \(\frac{1}{3}\)