Question

consider a cone with a height of 9 feet and a volume of 48π cubic feet. 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: Write 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 know that \(V = 48\pi\) and \(h=9\). Substituting these into the formula gives \(48\pi=\frac{1}{3}\pi r^{2}\times9\).

Step3: Simplify the equation

First, simplify the right - hand side: \(\frac{1}{3}\times9 = 3\), so the equation becomes \(48\pi=3\pi r^{2}\).
Then, divide both sides by \(3\pi\): \(\frac{48\pi}{3\pi}=r^{2}\). Since \(\frac{48\pi}{3\pi}=16\), we have \(r^{2}=16\).

Step4: Solve for \(r\)

Take the square root of both sides. Since \(r>0\) (radius cannot be negative), \(r = \sqrt{16}\).

Answer:

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