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: Recall volume formulas

The volume of a cylinder $V_{cylinder}=\pi r_{1}^{2}h_{1}$, and the volume of a cone $V_{cone}=\frac{1}{3}\pi r_{2}^{2}h_{2}$. Given $V_{cylinder} = V_{cone}$, $r_{1}=6$ inches, $h_{1}=6$ inches, and $r_{2}=12$ inches.

Step2: Set up the equation

$\pi r_{1}^{2}h_{1}=\frac{1}{3}\pi r_{2}^{2}h_{2}$. Cancel out $\pi$ on both sides of the equation, we get $r_{1}^{2}h_{1}=\frac{1}{3}r_{2}^{2}h_{2}$.

Step3: Substitute the values

Substitute $r_{1} = 6$, $h_{1}=6$, and $r_{2}=12$ into the equation: $6^{2}\times6=\frac{1}{3}\times12^{2}\times h_{2}$.
First, calculate $6^{2}\times6=36\times6 = 216$ and $12^{2}=144$. The equation becomes $216=\frac{1}{3}\times144\times h_{2}$.

Step4: Solve for $h_{2}$

$\frac{1}{3}\times144\times h_{2}=216$, $48h_{2}=216$, then $h_{2}=\frac{216}{48}= 4.5$ inches. But there is a mistake above. Redo step 3 correctly:
Substitute $r_{1} = 8$, $h_{1}=6$, and $r_{2}=12$ into $r_{1}^{2}h_{1}=\frac{1}{3}r_{2}^{2}h_{2}$.
We have $8^{2}\times6=\frac{1}{3}\times12^{2}\times h_{2}$.
$64\times6=\frac{1}{3}\times144\times h_{2}$.
$384 = 48h_{2}$.

Step5: Find the height of the cone

Solve for $h_{2}$: $h_{2}=\frac{384}{48}=8$ inches.

Answer:

B. 8 inches