The diameter and height of this cylinder are equal to the side length, S, of the cube in which the cylinder is inscribed. What is the expression for the cylinder's volume?
The volume of a cylinder can be calculated using the formula V = πr^2h, where r is the radius of the cylinder and h is the height of the cylinder.
In this case, the diameter of the cylinder is equal to the side length of the inscribed cube, which means the radius is half of the side length.
Therefore, the expression for the cylinder's volume is V = π(S/2)^2h.
To find the expression for the cylinder's volume, we need to know the formula for the volume of a cylinder. The formula for the volume of a cylinder is given by:
V = πr^2h
where V represents the volume, r represents the radius of the base, and h represents the height of the cylinder.
In this case, the diameter of the cylinder is equal to the side length, S, of the cube in which the cylinder is inscribed. The diameter is twice the radius, so we can say that the radius, r, is equal to S/2.
Since the height of the cylinder is also equal to the side length, S, of the cube, we can say that h = S.
Substituting these values into the formula for the volume of a cylinder:
V = π(S/2)^2(S)
Simplifying further:
V = π(S^2/4)(S)
V = (πS^3)/4
Therefore, the expression for the volume of the cylinder is (πS^3)/4.