Question

a cone has a volume of 144π and a height of 12. what is the radius of the base?

Explanation:

Step1: Write the formula for the volume of a cone

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

Step2: Substitute the given values into the formula

Given \(V = 144\pi\) and \(h=12\), substitute into \(V=\frac{1}{3}\pi r^{2}h\):
\(144\pi=\frac{1}{3}\pi r^{2}\times12\)

Step3: Simplify the equation

First, cancel out \(\pi\) on both sides of the equation. Then simplify \(\frac{1}{3}\times12 = 4\). The equation becomes \(144 = 4r^{2}\).

Step4: Solve for \(r^{2}\)

Divide both sides of \(144 = 4r^{2}\) by 4: \(r^{2}=\frac{144}{4}=36\).

Step5: Solve for \(r\)

Take the square root of both sides. Since \(r>0\) (radius is a non - negative quantity), \(r=\sqrt{36}=6\).

Answer:

\(6\)