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) This is bcause I have already worked out the answers to a) and b)
How did you work out parts a) and b)
see your other posts on this
a)area of sector= x over 360* pie r squared
40 over 360*pie*12 squared
=50.3cm squared
b) area of base= 320 over 360*pie* 12 squared= 402.1238597
vol of prism = area of base* height
= 402.1238597*10= 4021cm cubed
by the way the top of the cake when the cake is sliced is 320 degrees
i know its listed as anonomous above but its me really so ive given you basically my how i worked out a and b including the vital answers.
To calculate the surface area of the cake that remains after the slice has been removed, we need to find the lateral surface area of the cylinder and subtract the area of the slice that has been cut out.
The lateral surface area of the cylinder can be calculated using the formula:
Lateral Surface Area = 2πrh
where r is the radius of the cylinder and h is the height of the cylinder.
In this case, the radius (r) of the cake is given as 12 cm and the height (h) of the cake is given as 10 cm. So, we can plug in these values into the formula to calculate the lateral surface area of the whole cake.
Lateral Surface Area of the Cake = 2π(12 cm)(10 cm)
= 240π cm²
Now, we need to subtract the area of the slice that has been cut out.
The area of a sector of a circle can be calculated using the formula:
Area of Sector = (θ/360°) x πr²
where θ is the angle of the sector in degrees and r is the radius of the circle.
The angle of the sector is not given directly, but we can calculate it using the properties of the circle and the cylinder. Since the top of the slice matches the shape of the top of the cake, the angle of the sector is the same as the angle formed by the missing slice.
To find the angle, we need to first calculate the circumference of the top of the cake, which is the same as the circumference of the whole cake.
Circumference of the Cake = 2πr
= 2π(12 cm)
= 24π cm
The length of the missing slice can be calculated using the formula for arc length:
Arc Length = (θ/360°) x Circumference
Given that the length of the slice is 1/4th of the circumference of the whole cake (since the top of the slice is a quarter of the circle), we can find the angle of the sector:
(θ/360°) x 24π = 1/4 x 24π
Simplifying the equation, we have:
(θ/360°) = 1/4
Now, we can solve for θ:
θ = 360° x (1/4)
= 90°
Therefore, the angle of the sector is 90°.
Now, we can calculate the area of the sector using the formula for the area of a sector:
Area of Sector = (θ/360°) x πr²
= (90°/360°) x π(12 cm)²
= (1/4) x π(12²) cm²
= 36π cm²
Finally, we can subtract the area of the sector from the lateral surface area to get the surface area of the cake that remains after the slice has been removed:
Surface Area of the Cake that Remains = Lateral Surface Area - Area of Sector
= 240π cm² - 36π cm²
= 204π cm²
Thus, the surface area of the cake that remains after the slice has been removed is 204π cm².