Question

consider a cone with a height of 6 inches and a volume of 32π cubic inches. find the exact radius of the cone. be sure to use the correct unit in your answer. the radius of the cone is

Explanation:

Step1: Recall the volume formula 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, and \(h\) is the height.

Step2: Substitute the given values into the formula

We are given that \(V = 32\pi\) in³ and \(h=6\) in. Substituting these into the formula \(V=\frac{1}{3}\pi r^{2}h\), we get \(32\pi=\frac{1}{3}\pi r^{2}\times6\).

Step3: Simplify the equation

First, simplify the right - hand side of the equation \(\frac{1}{3}\pi r^{2}\times6 = 2\pi r^{2}\). So the equation becomes \(32\pi=2\pi r^{2}\).
Divide both sides of the equation by \(2\pi\). We have \(\frac{32\pi}{2\pi}=r^{2}\), which simplifies to \(r^{2} = 16\).

Step4: Solve for \(r\)

Take the square root of both sides. Since \(r>0\) (radius is a non - negative quantity), \(r=\sqrt{16}=4\).

Answer:

\(4\) in