Question

find the volume of a cone with a base diameter of 12 m and a height of 9 m. write the exact volume in terms of π, and be sure to include the correct unit in your answer.

Explanation:

Step1: Find the radius of the base

The diameter of the base is 12 m, so the radius \( r \) is half of the diameter.
\( r = \frac{12}{2} = 6 \, \text{m} \)

Step2: 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.

Step3: Substitute the values of \( r \) and \( h \) into the formula

We know \( r = 6 \, \text{m} \) and \( h = 9 \, \text{m} \). Substituting these values into the formula:
\( V = \frac{1}{3}\pi (6)^2 (9) \)

Step4: Simplify the expression

First, calculate \( (6)^2 = 36 \). Then:
\( V = \frac{1}{3}\pi \times 36 \times 9 \)
Simplify \( \frac{1}{3} \times 36 = 12 \), then:
\( V = 12 \times 9 \times \pi \)
\( V = 108\pi \)
The unit for volume is cubic meters (\( \text{m}^3 \)).

Answer:

\( 108\pi \, \text{cubic meters} \) or \( 108\pi \, \text{m}^3 \)