Question

a cylinder and a cone are shown. which explains whether the bases of the cylinder and the cone have the same area? the bases have the same area because the volume of the cone is \\( \frac { 1 } { 2 } \\) the volume of the cylinder. the bases have the same area because the heights are the same. the bases do not have the same area because the volumes are not the same. the bases do not have the same area because the volume of the cylinder is not 3 times the volume of the cone, given the same heights.

Explanation:

Step1: Recall volume formulas

The volume formula for a cylinder is \( V_{cylinder}=B_{cylinder}h_{cylinder}\), and for a cone is \( V_{cone}=\frac{1}{3}B_{cone}h_{cone}\). Given \(h_{cylinder} = h_{cone}=12\) inches.

Step2: Substitute values into formulas

For the cylinder: \(2512 = B_{cylinder}\times12\), so \(B_{cylinder}=\frac{2512}{12}\). For the cone: \(1256=\frac{1}{3}B_{cone}\times12\), then \(1256 = 4B_{cone}\), and \(B_{cone}=\frac{1256}{4}=314\). Also, if we use the relationship \(V_{cylinder}=3V_{cone}\) (when \(h_{cylinder} = h_{cone}\)), check \(3\times1256 = 3768
eq2512\).

Answer:

The bases do not have the same area because the volume of the cylinder is not 3 times the volume of the cone, given the same heights.