Question

1. which of the following best explains why a cones volume is \\( \frac { 1 } { 3 } \\) that of a cylinder?

• a. a cone occupies one - third of the space of a cylinder with the same base and height.
• b. a cone has a smaller base area than a cylinder.
• c. a cones height is always one - third of its radius.
• d. a cone is shorter than a cylinder with the same radius and height.

Explanation:

The volume formula for a cylinder is \(V_{cylinder}=\pi r^{2}h\) (where \(r\) is the radius of the base and \(h\) is the height). The volume formula for a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h\). When a cone and a cylinder have the same base (so same \(r\)) and same height (\(h\)), by comparing the two formulas, we can see that the cone's volume is one - third of the cylinder's volume. This means a cone occupies one - third of the space of a cylinder with the same base and height.
Option b is incorrect because if they have the same base (same radius for circular bases), the base areas (\(\pi r^{2}\)) are equal. Option c is incorrect as there is no such general relationship that a cone's height is one - third of its radius. Option d is incorrect because when we talk about the volume relationship \(\frac{V_{cone}}{V_{cylinder}}=\frac{1}{3}\) (for same \(r\) and \(h\)), it's not about the cone being shorter (they have the same height in the volume - comparison context).

Answer:

A. A cone occupies one - third of the space of a cylinder with the same base and height.