Question

1. a cylinder and a cone have the same volume and base radius. what must be true about their heights?

a. the height of the cone is one - third the height of the cylinder.
b. the height of the cone is equal to the height of the cylinder.
c. the height of the cone is twice the height of the cylinder.
d. the height of the cone is three times the height of the cylinder.

Explanation:

Step1: Write the volume formulas

The volume formula for a cylinder is \(V_{cylinder}=\pi r^{2}h_{cylinder}\), and for a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h_{cone}\).

Step2: Set the volumes equal

Since \(V_{cylinder} = V_{cone}\) and \(r_{cylinder}=r_{cone}\), we have \(\pi r^{2}h_{cylinder}=\frac{1}{3}\pi r^{2}h_{cone}\).

Step3: Solve for the relationship between heights

Cancel out \(\pi r^{2}\) from both sides. We get \(h_{cylinder}=\frac{1}{3}h_{cone}\), which can be rewritten as \(h_{cone} = 3h_{cylinder}\).

Answer:

d. The height of the cone is three times the height of the cylinder.