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)
In terms of surface area, does that refer to the remaining frosted area, or does it include the internal area where the slice was removed? (I assume that you will not include the surface area resting on the plate.)
If the former, subtract the area of top of slice from total top area. Use the portion of the remaining circumference times the height for the area of the sides.
If the latter, add 2 times radius times height to the above area.
To calculate the surface area of the cake that remains after the slice has been removed, you need to consider two parts: the curved surface area of the outer surface of the cake and the top and bottom surfaces of the remaining cake.
Let's break down the calculations step by step:
1. Curved Surface Area:
The curved surface area of a cylinder is given by the formula A = 2πrh, where A is the curved surface area, π is a constant (approximately 3.14159), r is the radius of the cylinder, and h is the height of the cylinder.
In this case, the radius of the cake is 12 cm, and the height is 10 cm. Thus, the curved surface area of the entire cake is A = 2π(12)(10) = 240π cm².
However, a sector of the top has been sliced out, which accounts for a portion of the curved surface area. To calculate the surface area of the slice, you need to find the length of the curved edge of the sector.
The total angle of the sector is 360°, but the angle of the slice that remains is 320°. To find the length of the curved edge of the sector, you can use the formula:
Length of curved edge = (angle/360) × circumference
The circumference of the top of the cake can be calculated using the formula: circumference = 2πr. In this case, the radius is 12 cm, so the circumference is 2π(12) = 24π cm.
Using the formula, the length of the curved edge of the sector is (320/360) × 24π = 8/9 × 24π = 64π/3 cm.
Therefore, the curved surface area of the remaining cake is 240π - 64π/3 cm².
2. Top and Bottom Surfaces:
The remaining cake still has the top and bottom surfaces intact. The top and bottom surfaces are both circular with a radius of 12 cm.
The area of a circle is given by the formula A = πr². Consequently, the total area of the top and bottom surfaces of the remaining cake is 2π(12)² = 288π cm².
3. Summing the Areas:
Next, you need to add the curved surface area and the area of the top and bottom surfaces to get the total surface area of the remaining cake.
Total surface area = Curved Surface Area + Top Surface Area + Bottom Surface Area
Total surface area = (240π - 64π/3) + 288π
Total surface area = 240π + 288π - 64π/3
Total surface area = (720π + 864π - 64π)/3
Total surface area = 1520π/3 cm²
Therefore, the surface area of the cake that remains after the slice has been removed is equal to 1520π/3 cm².