Question

complete the steps to find the surface area of the cylinder. the area of the base is (pi) in.(^{2}) the circumference of the base is (pi) in. the area of the lateral surface is (pi) in.(^{2}) the surface area is (pi) in.(^{2})

Explanation:

Step1: Calculate the area of the base

The formula for the area of a circle (base of the cylinder) is \(A = \pi r^{2}\). Given \(r = 3\) in, then \(A=\pi\times3^{2}=9\pi\) in\(^{2}\).

Step2: Calculate the circumference of the base

The formula for the circumference of a circle is \(C = 2\pi r\). Given \(r = 3\) in, then \(C = 2\pi\times3=6\pi\) in.

Step3: Calculate the lateral - surface area

The formula for the lateral - surface area of a cylinder is \(A_{l}=Ch\) (where \(C\) is the circumference of the base and \(h\) is the height of the cylinder). Given \(C = 6\pi\) in and \(h = 4\) in, then \(A_{l}=6\pi\times4 = 24\pi\) in\(^{2}\).

Step4: Calculate the surface area

The formula for the surface area of a cylinder is \(A_{s}=2A+A_{l}\) (where \(A\) is the area of the base and \(A_{l}\) is the lateral - surface area). Since \(A = 9\pi\) in\(^{2}\) and \(A_{l}=24\pi\) in\(^{2}\), then \(A_{s}=2\times9\pi+24\pi=18\pi + 24\pi=42\pi\) in\(^{2}\).

Answer:

The area of the base is \(9\pi\) in\(^{2}\). The circumference of the base is \(6\pi\) in. The area of the lateral surface is \(24\pi\) in\(^{2}\). The surface area is \(42\pi\) in\(^{2}\).