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.
thankyou, I wanted to know whether I was right or the textbook because it was very rare that they make mistakes, I wanted to clarify whether it makes sense to add the 2 sides area=240. I thought it did so Im going to leave it how I did it.
To calculate the surface area of the cake that remains after the slice has been removed, you need to consider the top, bottom, and sides of the cake.
First, calculate the area of the top and bottom of the cake:
Area of the top (sector) = (320/360) * π * (12)^2
Area of the bottom (circle) = π * (12)^2
Next, calculate the area of the curved side of the cake (excluding the slice):
Circumference of the cake = 2 * π * 12
Height of the cake = 10
Area of the curved side = Circumference * Height = 2 * π * 12 * 10
Finally, calculate the total surface area of the remaining cake by adding all these areas together:
Total surface area = Area of top + Area of bottom + Area of curved side
Substituting the values:
Total surface area = [(320/360) * π * (12)^2] + [π * (12)^2] + [2 * π * 12 * 10]
Simplifying the expression, you should get the correct answer.
To calculate the surface area of the cake that remains after the slice has been removed, you need to consider the area of the base, the lateral surface area, and the top surface area.
1) Area of the base:
To calculate the area of the base, you used the formula for the area of a sector of a circle. You correctly calculated it as 320/360 * π * 12^2 = 804.2477193 square cm. However, since there is a top and bottom to the cake, you need to multiply this value by 2, resulting in a total of 1608.4954386 square cm.
2) Lateral Surface Area:
To calculate the lateral surface area, you used the formula for the rectangle's surface area. You correctly calculated it as 320/360 * 2 * π * 12 * 10 = 670.2064328 square cm.
3) Side areas:
You mentioned that there are two rectangular sides along the circumference of the cake. Here, while you considered the lateral surface area, you didn't consider the area of the side of the slice that has been cut out. The area of the side of the slice can be calculated as the perimeter of the sector multiplied by the height of the slice. The perimeter of the sector is (320/360) * 2 *π * 12 = 67.02064328 cm. Multiplying this by the height of the cake (10 cm), you get an additional area of 670.2064328 square cm.
However, it seems like you made an error when adding these areas together. Instead of 240 + 804.2477193 + 670.2064328, the correct calculation should be 1608.4954386 + 670.2064328 + 240 = 2518.7028714 square cm.
So, the correct answer for the surface area of the remaining cake is 2518.7028714 square cm, not 1474 cm² as mentioned in the book.