Question

the formula for surface area for a cylinder can be manipulated and applied to cake preparation to determine the amount of frosting needed to cover the entirety of the cake. the full formula is as follows: surface area = 2πrh + 2πr² however, 2πr² assumes both the top and the bottom of the cylinder will be used in calculation, but for a cake only the top needs frosting. manipulate the formula as necessary and solve for a cake that will be 5 inches tall with a diameter of 6 inches. use the value of 3.14 for pi. how many square inches will the frosting need to cover? round to the nearest hundredth. you do not need to include a unit label.

Explanation:

Step1: Find the radius

Given diameter \(d = 6\) inches, radius \(r=\frac{d}{2}\).
\(r=\frac{6}{2}=3\) inches.

Step2: Adjust the formula

For a cake, the formula for the surface area (frosting area) is \(A = 2\pi rh+\pi r^{2}\) (since only one circular face - the top - is frosted).

Step3: Substitute values

Substitute \(r = 3\), \(h = 5\), and \(\pi=3.14\) into \(A = 2\pi rh+\pi r^{2}\).
First term: \(2\pi rh=2\times3.14\times3\times5\).
\(2\times3.14\times3\times5=(2\times3\times5)\times3.14 = 30\times3.14=94.2\).
Second term: \(\pi r^{2}=3.14\times3^{2}\).
\(3.14\times3^{2}=3.14\times9 = 28.26\).

Step4: Calculate the total

\(A=94.2 + 28.26\).
\(A=122.46\).

Answer:

122.46