Question

examine the composite solid shown below. the volume of the cylinder is 1296πin³, and the volume of the composite solid is 1728πin³. which of the following statements is true? 4 of 9 questions the height of the cone is three times the height of the cylinder. the height of the cylinder is the same as the height of the cone. the height of the cylinder is three times the height of the cone. the height of the cylinder is two times the height of the cone.

Explanation:

Step1: Calculate the volume of the cone

The volume of the composite solid \(V_{composite}=V_{cylinder}+V_{cone}\). Given \(V_{cylinder} = 1296\pi\) and \(V_{composite}=1728\pi\), then \(V_{cone}=V_{composite}-V_{cylinder}\).

$$V_{cone}=1728\pi - 1296\pi=432\pi$$

Step2: Use the volume formula for cylinder and cone

The volume formula for a cylinder is \(V_{cylinder}=\pi r^{2}h_{cylinder}\), and for a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h_{cone}\). Given \(r = 12\) (radius of both cylinder and cone as they are part of the composite solid), from \(V_{cylinder}=\pi\times12^{2}\times h_{cylinder}=1296\pi\), we can solve for \(h_{cylinder}\):

$$144\pi h_{cylinder}=1296\pi\Rightarrow h_{cylinder}=\frac{1296\pi}{144\pi}=9$$

From \(V_{cone}=\frac{1}{3}\pi\times12^{2}\times h_{cone}=432\pi\), we can solve for \(h_{cone}\):

$$\frac{1}{3}\times144\pi h_{cone}=432\pi\Rightarrow48\pi h_{cone}=432\pi\Rightarrow h_{cone}=\frac{432\pi}{48\pi}=9$$

Answer:

The height of the cylinder is the same as the height of the cone.