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 because I have already worked out the answers to a) and b)
c)
I worked out the area of the cake using the sector 320 degrees, but there is also a bottom so you therefore have to multiple it by 2.
same with when you cut the slice- there are 2 rectangular sides. obviously round the circumference of the cake there is a long rectangular shape but because it is a cake it looks curved. because you want the whole surface area once you've sliced it you have to add all the figures. Here are my workings:
10*12=120*2=240cm squared
area of base=320 over 360*pie*12 squared=402.1238597*2 cos theres a top and bottom 4 a cake= 804.2477193
w=320 over 360*2*pie*12 ( multiply this by height of 10cm= 670.2064328
add these: 240+804.2477193+670.2064328=1714.454852 so surface area = 1414cm squared
however in the book the answer was 1474cm squared. i figured out they didn't add the 240 on which are the bits on the side- which is right?
It's me again.
I now fully understand what you are asking for in your part c)
I agree with your calculations of the exposed area of the top and bottom of the remaining cake
area (top+bottom) = 2(320/360)(144π) = 804.25 (you had that)
area (exposed outer side) = (320/360)(2π(12))(10) = 670.21 (you had that also)
if we add just those two sums we get 1474.45 which is their answer.
So it does look like they forgot the 2 rectangular areas of the inside of the remaining cake, which you calculated correctly as well, which were 2(12)(10) = 240
so the correct answer should have been the sum of those 3 answers or 1714.45 which you also had.
I again apologize for my error in our last exchange.
that is totally wrong
To calculate the surface area of the cake that remains after the slice has been removed, you need to consider all the different surfaces of the cake.
1. The area of the top and bottom surfaces of the cake:
You correctly calculated the area of the top surface using the sector angle of 320 degrees. The formula for the area of a sector is (angle/360) * π * r^2, where r is the radius. However, since there are two surfaces (top and bottom), you need to multiply this result by 2.
Area of top and bottom surfaces = 2 * (320/360) * π * 12^2
2. The area of the curved surface of the cake:
The curved surface of the cake forms a rectangle when it is unrolled. You need to calculate the length of this rectangle. The length of the rectangle is equal to the circumference of the circular base, which is given by the formula 2 * π * r. Multiply this circumference by the height of the cake to get the area of the curved surface.
Area of curved surface = (320/360) * 2 * π * 12 * 10
3. The area of the sides of the cake:
The sides of the cake, excluding the curved surface, form two rectangles. Each rectangle's area is equal to the height of the cake multiplied by the length of the sector's arc. The length of the sector's arc can be calculated using the formula (angle/360) * 2 * π * r.
Area of side surfaces = 2 * 10 * (320/360) * 2 * π * 12
Now you can calculate the total surface area:
Total surface area = Area of top and bottom surfaces + Area of curved surface + Area of side surfaces
Plug in the correct formulas and values to get the final answer. Double-check your calculations to make sure you haven't made any errors.