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.
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)
You will have to tell us something about the sector cut out, either its sector angle or its arc length.
sector angle is 320 degrees without sliced cut which is therefore 40 degrees
To calculate the surface area of the cake that remains after the slice has been removed, you first need to find the lateral surface area of the cake (excluding the top and bottom circles). Then you will subtract the area of the slice that was removed.
Here's how you can calculate it step by step:
Step 1: Calculate the lateral surface area of the cake:
The formula for the lateral surface area of a cylinder is given by: Lateral Surface Area = 2 * π * r * h, where r is the radius and h is the height of the cylindrical cake.
In this case, the radius (r) is given as 12 cm, and the height (h) is given as 10 cm.
So the lateral surface area of the cake is: A_lateral = 2 * π * 12 * 10 = 240π cm².
Step 2: Calculate the area of the slice that was removed:
The shape of the top of the slice is a sector of the circle. The area of a sector can be calculated using the formula: Area = (θ/360) * π * r², where θ is the central angle in degrees and r is the radius of the circle.
Since the shape of the top of the slice is a sector of the circle, we need to find the central angle (θ) of the sector. To do this, we can use the fact that the height of the cake (10 cm) is also the height of the sector. The central angle can be found using the formula: θ = (h/r) * 360.
Plugging in the values, we get: θ = (10/12) * 360 ≈ 300°.
Now we can calculate the area of the slice as: A_slice = (300/360) * π * 12².
Step 3: Calculate the surface area of the remaining cake:
To find the surface area of the remaining cake, subtract the area of the slice from the lateral surface area: A_remaining = A_lateral - A_slice.
A_remaining ≈ 240π - ((300/360) * π * 12²).
Simplifying the expression, we get: A_remaining ≈ 240π - (0.8333 * π * 12²).
Evaluating the expression, we get the surface area of the remaining cake.
Please note that the result will be in square units (cm²) since we are measuring surface area.