Question

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

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

Explanation:

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 \(V_{cone}=\frac{1}{3}\pi r^{2}h\). When a cone and a cylinder have the same base (so same radius \(r\)) and same height \(h\), by comparing the two formulas, we can see that the volume of the cone is one - third of the volume of the cylinder.

• Option A: The height of a cone is not related to its radius in the way described for the volume relationship.
• Option B: Just being shorter (without considering base area) is not a valid reason for the volume relationship.
• Option C: If they have the same base (same radius), their base areas (\(A = \pi r^{2}\)) are equal.

Answer:

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