a cylindrical cake with radius 12cm and height 10cm has a slice cut out. The shape of the top of the slice is a sector of the circle that forms the top of the cake. Exluding the sliced piece, the angle is 320 degrees.

a) Calculate the area on top of the slice that has been cut out.
b) Calculate the volume of the cake that remains after the slice has been removed.
c) Calculate the surface area of the cake that remains after the slice has been removed.

i particularly need to know how to do c) This is bcause I have already worked out the answers to a) and b)

To calculate the surface area of the cake that remains after the slice has been removed, we need to consider the lateral surface area of the cylindrical shape and subtract the area of the sliced sector.

a) To calculate the area on top of the slice that has been cut out, we need to find the area of the sector. The formula for the area of a sector is (θ/360) * πr^2, where θ is the angle in degrees and r is the radius of the circle. Plugging in the values, we have (320/360) * π * 12^2 = (8/9) * π * 144 ≈ 402.12 cm^2.

b) To calculate the volume of the cake that remains after the slice has been removed, we need to find the volume of the cylinder and subtract the volume of the sliced sector. The formula for the volume of a cylinder is πr^2h, where r is the radius and h is the height of the cylinder. Plugging in the values, we have π * 12^2 * 10 = π * 144 * 10 ≈ 4523.89 cm^3. Since the volume of the sliced sector is already calculated in the previous step, we subtract it from the total volume to get the remaining volume: 4523.89 - 402.12 ≈ 4121.77 cm^3.

c) Now, to calculate the surface area of the cake that remains after the slice has been removed, we need to find the lateral surface area of the cylinder and subtract the area of the sliced sector. The formula for the lateral surface area of a cylinder is 2πrh, where r is the radius and h is the height of the cylinder. Plugging in the values, we have 2 * π * 12 * 10 = 240π cm^2. Since the area of the sliced sector is already calculated in part a), we subtract it from the lateral surface area to get the remaining surface area: 240π - 402.12 ≈ 694.77 cm^2.

Therefore, the surface area of the cake that remains after the slice has been removed is approximately 694.77 cm^2.