Question

a cylinder and a cone have the same volume. the cylinder has a radius of 8 inches and a height of 6 inches. the cone has a radius of 12 inches. what is the height of the cone? a. 4 inches b. 8 inches c. 9 inches d. 10 inches e. 12 inches

Explanation:

Step1: Find volume of the cylinder

The volume formula for a cylinder is $V_{cylinder}=\pi r_{1}^{2}h_{1}$. Given $r_{1} = 8$ inches and $h_{1}=6$ inches. So $V_{cylinder}=\pi\times8^{2}\times6=\pi\times64\times6 = 384\pi$ cubic - inches.

Step2: Set up the volume formula for the cone

The volume formula for a cone is $V_{cone}=\frac{1}{3}\pi r_{2}^{2}h_{2}$. We know $V_{cone}=V_{cylinder}=384\pi$ and $r_{2} = 12$ inches. Substitute these values into the cone - volume formula: $384\pi=\frac{1}{3}\pi\times12^{2}\times h_{2}$.

Step3: Solve for the height of the cone

First, simplify the right - hand side of the equation. $\frac{1}{3}\pi\times12^{2}\times h_{2}=\frac{1}{3}\pi\times144\times h_{2}=48\pi h_{2}$. So the equation becomes $384\pi = 48\pi h_{2}$. Divide both sides of the equation by $48\pi$. $\frac{384\pi}{48\pi}=h_{2}$, and $h_{2}=8$ inches.

Answer:

B. 8 inches