Question

determine if each statement is always, sometimes, or never true.

	always	sometimes	never
a cylinder and a cone with the same height have the same volume.
a cylinder and a cone with the same radius and the same height have the same volume.

Explanation:

Step1: Recall volume formulas

The volume formula for a cylinder is \(V_{cylinder}=\pi r^{2}h\). The volume formula for a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h\).

Step2: Analyze first statement

Let \(r_{1} = r_{2}=r\). \(V_{cylinder}=\pi r^{2}h_{1}\), \(V_{cone}=\frac{1}{3}\pi r^{2}h_{2}\). If \(h_{2} = 3h_{1}\), then \(V_{cylinder}=V_{cone}\). So a cylinder and a cone with the same radius can have the same volume.

Step3: Analyze second statement

Let \(h_{1}=h_{2} = h\). \(V_{cylinder}=\pi r_{1}^{2}h\), \(V_{cone}=\frac{1}{3}\pi r_{2}^{2}h\). If \(r_{1}^{2}=\frac{1}{3}r_{2}^{2}\) (i.e., \(r_{1}=\frac{1}{\sqrt{3}}r_{2}\)), then \(V_{cylinder}=V_{cone}\). So a cylinder and a cone with the same height can have the same volume.

Step4: Analyze third statement

Let \(r_{1}=r_{2}=r\) and \(h_{1}=h_{2}=h\). \(V_{cylinder}=\pi r^{2}h\), \(V_{cone}=\frac{1}{3}\pi r^{2}h\). Since \(\pi r^{2}h
eq\frac{1}{3}\pi r^{2}h\) (unless \(h = 0\), but for non - degenerate solids \(h>0\)), a cylinder and a cone with the same radius and height never have the same volume.

Answer:

1. Sometimes
2. Sometimes
3. Never