Question

find the volume of a cone with a base radius of 6 m and a height of 10 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 formula for the volume \( V \) of a cone is \( V=\frac{1}{3}\pi r^{2}h \), where \( r \) is the radius of the base and \( h \) is the height of the cone.

Step2: Identify the values of \( r \) and \( h \)

From the problem, the radius \( r = 6\space m \) and the height \( h=10\space m \).

Step3: Substitute the values into the formula

Substitute \( r = 6 \) and \( h = 10 \) into the formula \( V=\frac{1}{3}\pi r^{2}h \). First, calculate \( r^{2} \): \( r^{2}=6^{2}=36 \). Then, multiply by \( h \) and \( \frac{1}{3}\pi \): \( V=\frac{1}{3}\pi\times36\times 10 \).

Step4: Simplify the expression

Simplify \( \frac{1}{3}\times36 = 12 \), then \( 12\times10=120 \). So the volume \( V = 120\pi\space m^{3} \).

Answer:

\( 120\pi \) cubic meters (or \( 120\pi\space m^{3} \))