Question

consider a cone with a height of 6 centimeters and a volume of 32π cubic centimeters. find the exact radius of the cone. be sure to use the correct unit in your answer. the radius of the cone is 4/π cm.

Explanation:

Step1: Recall the volume formula for a cone

The volume formula for 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 \( V = 32\pi\) \(cm^{3}\) and \( h=6\) \(cm\). Substituting these into the formula gives \( 32\pi=\frac{1}{3}\pi r^{2}\times6\).

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 \(32\pi=2\pi r^{2}\).

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

Divide both sides of the equation by \(2\pi\): \(\frac{32\pi}{2\pi}=r^{2}\), which simplifies to \(r^{2} = 16\).

Step5: Solve for \(r\)

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

Answer:

The radius of the cone is \(4\) \(cm\).