Question

what can be concluded about a cylinder and a cone with the same height and radius measures? choose three correct answers.
the cone and cylinder have the same base area.
the cone has the greater volume.
the cone and cylinder have the same diameter.
the cylinder can fit inside the cone.
the volume of the cone is \\( \frac { 1 } { 3 } \\) the volume of the cylinder.

Explanation:

Step1: Volume formulas

The volume formula for a cylinder is \(V_{cylinder}=\pi r^{2}h\) (where \(r\) is the radius and \(h\) is the height). The volume formula for a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h\). If they have the same radius (\(r\)) and height (\(h\)), then:

• Since \(r\) is the same, the base area \(A = \pi r^{2}\) is the same for both (because base area of cylinder and cone with same - radius is calculated by the same formula \(A=\pi r^{2}\)). Also, if radius \(r\) is the same, diameter \(d = 2r\) is the same.
• From \(V_{cylinder}=\pi r^{2}h\) and \(V_{cone}=\frac{1}{3}\pi r^{2}h\), we can see that \(V_{cone}=\frac{1}{3}V_{cylinder}\)

Answer:

• The cone and cylinder have the same base area.
• The cone and cylinder have the same diameter.
• The volume of the cone is \(\frac{1}{3}\) the volume of the cylinder.