Question

1. a cylinder has radius 9 inches and height 15 inches. a cone has the same radius and height.

a. find the volume of the cylinder.
b. find the volume of the cone.
c. what fraction of the cylinders volume is the cones volume?
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Explanation:

Step1: Volume formula of cylinder

The volume formula of a cylinder is \(V_{cylinder}=\pi r^{2}h\). Given \(r = 9\) inches and \(h=15\) inches.
Substitute the values into the formula: \(V_{cylinder}=\pi\times9^{2}\times15\).
First calculate \(9^{2}=81\), then \(V_{cylinder}=\pi\times81\times15 = 1215\pi\approx1215\times3.14 = 3815.1\) cubic inches.

Step2: Volume formula of cone

The volume formula of a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h\). Since \(r = 9\) inches and \(h = 15\) inches.
Substitute the values into the formula: \(V_{cone}=\frac{1}{3}\pi\times9^{2}\times15\).
First calculate \(9^{2}=81\), then \(V_{cone}=\frac{1}{3}\pi\times81\times15=405\pi\approx405\times3.14 = 1271.7\) cubic inches.

Step3: Find the fraction

To find what fraction of the cylinder's volume is the cone's volume, use the formula \(\text{Fraction}=\frac{V_{cone}}{V_{cylinder}}\).
Since \(V_{cylinder}=1215\pi\) and \(V_{cone}=405\pi\), then \(\text{Fraction}=\frac{405\pi}{1215\pi}=\frac{1}{3}\)

Answer:

a. The volume of the cylinder is approximately \(3815.1\) cubic inches.
b. The volume of the cone is approximately \(1271.7\) cubic inches.
c. The cone's volume is \(\frac{1}{3}\) of the cylinder's volume.