Question

what is the relationship between the volume of a cone and the volume of a cylinder with the same height and radius?
the volume of the cone will be greater than that of the cylinder.
the volume of the cone will be one - third that of the cylinder.
the volume of the cone will be one - half that of the cylinder.
the volume of the cone will be the same as the cylinder.

Explanation:

Step1: Recall volume formulas

Volume of a cylinder \( V_{cylinder}=\pi r^{2}h\). Volume of a cone \( V_{cone}=\frac{1}{3}\pi r^{2}h\), where \( r \) is the radius and \( h \) is the height.

Step2: Compare the volumes

If we divide the volume of the cone by the volume of the cylinder (\(V_{cylinder}
eq0\)), we get \(\frac{V_{cone}}{V_{cylinder}}=\frac{\frac{1}{3}\pi r^{2}h}{\pi r^{2}h}\). The \(\pi r^{2}h\) terms cancel out, and \(\frac{V_{cone}}{V_{cylinder}}=\frac{1}{3}\).

Answer:

The volume of the cone will be one - third that of the cylinder.