Question

find the volume of a cone with a base radius of 6 m and a height of 6 m.
write the exact volume in terms of π, and be sure to include the correct unit in your answer.

Explanation:

Step1: Recall the formula for the volume of a cone

The volume \( V \) of a cone is given by the formula \( V=\frac{1}{3}\pi r^{2}h \), where \( r \) is the radius of the base and \( h \) is the height of the cone.

Step2: Substitute the given values into the formula

We are given that the radius \( r = 6\space m \) and the height \( h=6\space m \). Substituting these values into the formula, we get:
\( V=\frac{1}{3}\pi\times(6)^{2}\times6 \)
First, calculate \( (6)^{2}=36 \). Then the expression becomes \( V = \frac{1}{3}\pi\times36\times6 \)

Step3: Simplify the expression

\( \frac{1}{3}\times36 = 12 \), so now we have \( V=12\times6\times\pi \)
\( 12\times6 = 72 \), so \( V = 72\pi\space m^{3} \) (Wait, there is a mistake in the original provided answer. Let's recalculate correctly)
Wait, let's do the calculation again. \( r = 6 \), \( h = 6 \)
\( V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\times6^{2}\times6=\frac{1}{3}\pi\times36\times6 \)
\( \frac{1}{3}\times36 = 12 \), then \( 12\times6=72 \), so the correct volume is \( 72\pi\space m^{3} \)

Answer:

\( 72\pi \) cubic meters (or \( 72\pi\space m^{3} \))