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)

ahhh, I knew you left something out

So the slice cut out has a 60° central angle, so its surface area must be 1/6 of the whole circle surface area, or (1/6)π(144)
and the part remaining must be (5/6)(144)π = 120π

why is it a 60 degrees central angle? arent you supposed to break the cylindrical cake into parts: top and bottom, 2 rectangles where the slice has been cut inside, outside curved when opened out becomes long rectangle?? then work out the area of each part then add up. i want to know how you calculate the area of each of those parts.I know how to do a rectangle

I see it as a simple round cake where we cut out a slice (a sector) leaving 320° of cake left.

I have no idea where you are getting your interpretation. Since when are cakes cut into tops and bottoms?

You say you got a) and b)
Were there answers given to this question?

not tops and bottoms, i mean like you have to work out the area of the top 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?

no its leaving 40 degrees of the cake left, 320 is wot is there

To calculate the surface area of the cake that remains after the slice has been removed, you need to find the lateral surface area of the cylindrical cake and subtract the area of the sliced portion.

a) To calculate the area on top of the slice that has been cut out, you need to find the area of the sector at the top of the cake.

The formula for the area of a sector is A = (θ/360) * π * r^2, where θ is the central angle in degrees, and r is the radius.

Given that the radius of the cake is 12 cm and the angle of the sector is 320 degrees, the area of the sliced portion is:

A = (320/360) * π * (12^2) cm^2
A = (8/9) * π * 144 cm^2
A ≈ 402.12 cm^2 (rounded to two decimal places)

b) To calculate the volume of the cake that remains after the slice has been removed, you need to find the volume of the entire cylindrical cake and subtract the volume of the sliced portion.

The formula for the volume of a cylinder is V = π * r^2 * h, where r is the radius and h is the height.

Given that the radius of the cake is 12 cm and the height is 10 cm, the volume of the entire cake is:

V = π * (12^2) * 10 cm^3
V = π * 144 * 10 cm^3
V ≈ 4523.89 cm^3 (rounded to two decimal places)

To find the volume of the sliced portion, you use the formula for the volume of a frustum of a cone:

V_sliced = (θ/360) * π * (r1^2 + r2^2 + r1*r2) * h_sliced, where r1 and r2 are the radii of the two circular bases of the frustum, θ is the central angle in degrees, and h_sliced is the height of the sliced portion.

Since the top of the slice is a sector of the circular top of the cake, we can use the same angle and radius, and the height of the sliced portion would be equal to the height of the cake.

V_sliced = (320/360) * π * (12^2 + 0^2 + 0*r1) * 10 cm^3
V_sliced = (8/9) * π * (144) * 10 cm^3
V_sliced ≈ 4021.15 cm^3 (rounded to two decimal places)

The volume of the cake that remains after the slice has been removed is:

V_remain = V - V_sliced
V_remain ≈ 4523.89 - 4021.15 cm^3
V_remain ≈ 501.74 cm^3 (rounded to two decimal places)

c) To calculate the surface area of the cake that remains after the slice has been removed, you need to find the lateral surface area of the cylindrical cake and subtract the area of the sliced portion.

The formula for the lateral surface area of a cylinder is A_lateral = 2 * π * r * h, where r is the radius and h is the height.

Given that the radius of the cake is 12 cm and the height is 10 cm, the lateral surface area of the entire cake is:

A_lateral = 2 * π * 12 * 10 cm^2
A_lateral = 240 * π cm^2
A_lateral ≈ 753.98 cm^2 (rounded to two decimal places)

To find the surface area of the sliced portion, we need to find the slant height, which is the distance along the curved surface from one point on the sliced portion to another.

The slant height can be calculated using the formula for the circumference of the sector:

C = (θ/360) * 2 * π * r, where θ is the central angle in degrees and r is the radius.

C = (320/360) * 2 * π * 12 cm
C = (8/9) * 2 * π * 12 cm
C ≈ 67.03 cm (rounded to two decimal places)

Since the slant height is the hypotenuse of a right triangle, you can use the Pythagorean theorem to find the height of the triangle:

height = √(slant height^2 - radius^2)
height = √(67.03^2 - 12^2) cm
height ≈ 65.34 cm (rounded to two decimal places)

The surface area of the sliced portion is the sum of the area of the sector and the area of the triangle formed by the height and the radius:

A_sliced = (320/360) * π * (12^2) cm^2 + (1/2) * 2 * π * 12 * 65.34 cm^2
A_sliced ≈ 402.12 cm^2 + 2412.39 cm^2
A_sliced ≈ 2814.51 cm^2 (rounded to two decimal places)

The surface area of the cake that remains after the slice has been removed is:

A_remain = A_lateral - A_sliced
A_remain ≈ 753.98 cm^2 - 2814.51 cm^2
A_remain ≈ -2060.53 cm^2 (rounded to two decimal places)

Note that the negative result indicates that the sliced portion has a larger surface area than the remaining portion of the cake.