a cylinder with radius r and height 2r + 4 contains a cube with edge length r√(2), as shown. what fraction of the cylinder's volume is taken up by the cube (in simplified form)
can't see the picture, but it sounds like the cube lies flat on the bottom of the cylinder, and it touches all four corners on the cylinder. Then the ratio of volumes is
(r√2)^3/(pi r^2 (2r+4))
Plug in your value of r for a numeric value.
I couldn't add a picture, but the cylinder is laying on its side and the cube is in the middle.
the side of r√2 clearly indicates the cube's base diagonal is the diameter of the cylinder. So, geez, pretend the cylinder is standing up already. I think my result is correct. Just simplify it some.
I have a link for the pic but I can post it.
To solve this problem, we need to find the volume of both the cylinder and the cube, and calculate the ratio of the cube's volume to the cylinder's volume.
The volume of a cylinder can be calculated using the formula: V_cylinder = π * r^2 * h, where r is the radius of the cylinder and h is the height.
In this case, the radius of the cylinder is r, and the height is 2r + 4. So the volume of the cylinder is: V_cylinder = π * r^2 * (2r + 4).
The volume of a cube can be calculated using the formula: V_cube = side^3, where side is the length of one side of the cube.
In this case, the length of one side of the cube is r√2, so the volume of the cube is: V_cube = (r√2)^3 = 2√2 * r^3.
Now we can calculate the fraction of the cylinder's volume taken up by the cube by dividing the volume of the cube by the volume of the cylinder:
Fraction = V_cube / V_cylinder = (2√2 * r^3) / (π * r^2 * (2r + 4)).
Now, let's simplify this expression:
Fraction = (2√2 * r^3) / (π * r^2 * (2r + 4)).
= (2√2 * r * r^2) / (π * r^2 * (2r + 4)).
= (2√2 * r) / (π * (2r + 4)).
= (√2 * r) / (π * (r + 2)).
So, the fraction of the cylinder's volume taken up by the cube, in simplified form, is (√2 * r) / (π * (r + 2)).