Question

question the volume of a right cone is 264π units³. if its circumference measures 12π units, find its height.

Explanation:

Step1: Find the radius from the circumference

The formula for the circumference of a circle is \( C = 2\pi r \). Given \( C = 12\pi \), we solve for \( r \):
\( 12\pi = 2\pi r \)
Divide both sides by \( 2\pi \): \( r=\frac{12\pi}{2\pi}=6 \)

Step2: Recall the volume formula for a cone

The volume \( V \) of a right cone is given by \( V=\frac{1}{3}\pi r^{2}h \), where \( r \) is the radius and \( h \) is the height. We know \( V = 264\pi \) and \( r = 6 \), so substitute these values into the formula:
\( 264\pi=\frac{1}{3}\pi(6)^{2}h \)

Step3: Solve for \( h \)

First, simplify the right - hand side: \( (6)^{2}=36 \), so the equation becomes \( 264\pi=\frac{1}{3}\pi\times36h \)
Simplify \( \frac{1}{3}\times36 = 12 \), so \( 264\pi = 12\pi h \)
Divide both sides by \( 12\pi \): \( h=\frac{264\pi}{12\pi}=22 \)

Answer:

The height of the right cone is 22 units.