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 with a diameter of 6 inches that needs 142.88 square inches of frosting to cover it. use the value of 3.14 for pi. what is the cakes height in inches? 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 \(r=\frac{d}{2}=\frac{6}{2}=3\) inches.

Step2: Adjust the surface - area formula

The original formula \(SA = 2\pi rh+2\pi r^{2}\). For a cake, we only need one circular face (top), so the formula becomes \(SA=\pi r^{2}+2\pi rh\).

Step3: Substitute the known values into the formula

We know \(SA = 142.88\), \(r = 3\), and \(\pi=3.14\).
Substitute into \(142.88=3.14\times3^{2}+2\times3.14\times3\times h\).
First, calculate \(3.14\times3^{2}=3.14\times9 = 28.26\).
The equation becomes \(142.88=28.26 + 18.84h\).

Step4: Solve for \(h\)

Subtract \(28.26\) from both sides: \(142.88−28.26=18.84h\).
\(114.62 = 18.84h\).
Then \(h=\frac{114.62}{18.84}\approx6.08\).

Answer:

\(6.08\)