First, use the formula for the volume of a cylinder to find the volume of the whole block of cheese. Then, using the given central angle, think about what part of the whole block is the cheese that remains.
≈ 7.07 cubic inches
Practice makes perfect
We know that after a party a sector of a cylindrical block of cheese that remains has a central angle of 45 ^(∘). We want to find the volume of the cheese that remains. To do it, let's begin by finding the volume of the whole block of cheese. Because we know it was cylindrical, we will recall the formula for the volume of a cylinder.
V = B h
In this formula B is the area of the base of a cylinder and h is the height. We are told that the block of cheese is 2 inches thick, so h = 2. The base of a cylindrical block is a circle, so B is the area of a circle. Let's substitute the formula for the area of a circle for B in the formula for the volume of a cylinder.
V = π r^2h
In this formula r is the radius of the circle. We are told that the block of cheese has a 6 -inch diameter. This means that the radius r is equal to 3. Let's substitute h= 2 and r = 3 into the above formula to find the volume of the whole block of cheese.
Now, to find the volume of the cheese that remains we need to think about what part of the whole block it is. We know that the central angle is 45 ^(∘). The base of the whole block is a circle, and recall that a circle has 360 ^(∘). This means that the cheese that remains is 45 360 of the whole block. To find its volume, we can multiply the volume of the whole block by 45 360 .
V_(cheese that remains) = 45/360 * V_(whole block)
Let's substitute 56.55 for the volume of the whole block to find the volume of the cheese that remains.
V_(cheese that remains) = 45/360 * V_(whole block)
