Question

the surface area of a cylinder is 180π square centimeters. the radius of the cylinder is 6 centimeters. what is the lateral area of the cylinder? write an exact answer in terms of π. square centimeters

Explanation:

Step1: Recall the formula for the surface area of a cylinder

The surface area formula of a cylinder is \(S = 2\pi r^{2}+2\pi rh\), where \(r\) is the radius and \(h\) is the height. The lateral (curved) surface area formula is \(L = 2\pi rh\).
We know that \(S=180\pi\) and \(r = 6\). First, find the area of the two - circular bases. The area of a circle is \(A=\pi r^{2}\), so the area of two circular bases is \(2\pi r^{2}\). Substitute \(r = 6\) into \(2\pi r^{2}\): \(2\pi\times6^{2}=2\pi\times36 = 72\pi\).

Step2: Calculate the lateral surface area

Since \(S=2\pi r^{2}+L\) (where \(L\) is the lateral surface area), and \(S = 180\pi\), \(2\pi r^{2}=72\pi\).
We can solve for \(L\) using the equation \(L=S - 2\pi r^{2}\).
Substitute \(S = 180\pi\) and \(2\pi r^{2}=72\pi\) into the equation: \(L=180\pi-72\pi\).

Answer:

\(108\pi\) square centimeters