frosting a layer cake involves frosting the top, a middle layer and the outside. A typical cake has a 9 inch diameter and is 4 inches high. Calculate the surface area of the cake that is frosted

frosting on the top and the curved sides, so

π(9/2)^2 + π(9)(4) = ?

To calculate the surface area of the cake that is frosted, we need to find the areas of the top, middle layer, and the outside of the cake.

1. Top: The top of the cake is a circle, so we can calculate its area using the formula for the area of a circle, which is A = πr^2. Since the cake has a 9-inch diameter, the radius (r) would be half of that, which is 4.5 inches. Therefore, the area of the top is A = π(4.5)^2.

2. Middle layer: The middle layer is also a circle, but its area is subtracted from the top. Since the middle layer is not specified, we'll assume it has a diameter of 8 inches, which means a radius of 4 inches. So, the area of the middle layer is A = π(4)^2.

3. Outside: The outside of the cake is essentially a cylindrical surface. To calculate its surface area, we need to find the area of the curved portion and the area of the two circular bases combined. The curved portion's area can be found using the formula A = 2πrh, where r is the radius and h is the height of the cake. The area of the two circular bases can be calculated using the formula A = 2πr^2.
- Curved portion: A = 2π(4)(4) = 32π
- Base 1: A1 = π(4)^2 = 16π
- Base 2: A2 = π(4)^2 = 16π
Therefore, the area of the outside of the cake is A = 32π + 16π + 16π = 64π.

To calculate the total frosted surface area, we need to subtract the area of the middle layer from the top and add the area of the outside:
Total Frosted Surface Area = (A_top - A_middle) + A_outside
= (π(4.5)^2 - π(4)^2) + 64π
= π(20.25 - 16) + 64π
= 4.25π + 64π

So, the total surface area of the cake that is frosted is approximately 68.25π square inches.