Each layer of a three-layer cake is a cylinder with a height of 8.5 cm.
The bottom layer has a diameter of 24 cm. The middle layer has a diameter of 21.5 cm
The top layer has a diameter of 19 cm
The surface of the cake is frosted.
Calculate the surface area (to the nearest square centimeter) of the cake when it is frosted.
First, calculate the surface area of each cylinder layer:
Bottom layer:
Radius = 24 cm / 2 = 12 cm
Surface area = 2 * π * 12 * 8.5 + π * 12^2 = 2 * π * 102 + π * 144 = 204π + 144
Middle layer:
Radius = 21.5 cm / 2 = 10.75 cm
Surface area = 2 * π * 10.75 * 8.5 + π * 10.75^2 = 2 * π * 91.375 + π * 115.5625 = 182.75π + 115.5625
Top layer:
Radius = 19 cm / 2 = 9.5 cm
Surface area = 2 * π * 9.5 * 8.5 + π * 9.5^2 = 2 * π * 80.75 + π * 90.25 = 161.5π + 90.25
Now, add the surface areas of all three layers to find the total surface area:
Total surface area = 204π + 144 + 182.75π + 115.5625 + 161.5π + 90.25
Total surface area = 548.25π + 349.8125
Calculate the approximate value of the total surface area:
Total surface area ≈ 548.25 * 3.14159 + 349.8125
Total surface area ≈ 1723.743 + 349.8125
Total surface area ≈ 2073.5555
Therefore, the surface area of the frosted cake is approximately 2074 square centimeters.