Question

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

Explanation:

Step1: Volume formula of cylinder

The volume formula of a cylinder is \(V_{cylinder}=\pi r^{2}h\) (where \(r\) is the radius of the base and \(h\) is the height).

Step2: Volume formula of cone

The volume formula of a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h\) (where \(r\) is the radius of the base and \(h\) is the height).

Step3: Compare the two volumes

When a cone and a cylinder have the same base (i.e., same \(r\)) and same height (i.e., same \(h\)), by comparing \(V_{cone}=\frac{1}{3}\pi r^{2}h\) and \(V_{cylinder}=\pi r^{2}h\), we can see that \(V_{cone}=\frac{1}{3}V_{cylinder}\). This means a cone occupies one - third of the space of a cylinder with the same base and height.
For option a: The height of a cone is not related to its radius in the way described for the volume relationship.
For option b: If they have the same base (same radius), the base area (\(A = \pi r^{2}\)) is the same.
For option c: When we compare the volumes of a cone and a cylinder in the volume formula relationship, the height is the same in the formula comparison (not that the cone is shorter).

Answer:

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