a pyramid has a spuare base with side length 20.a right circular cylinder has a diameter of 10 and a length of 10.the cylinder is lying on its side,completely inside the pyramid.the central axis of the cylinder lies parallel to and directly above a diagonal of the pyramids base.the midpoint of the central axis lies directly above the centre of the square base of the pyramid
And the question is?
To find the volume of the space inside the pyramid that is not occupied by the cylinder, we need to first find the volume of the pyramid and the volume of the cylinder.
Volume of a pyramid:
The volume of a pyramid is given by the formula V = (1/3) * base area * height.
In this case, the base of the pyramid is a square with a side length of 20. So, the base area is calculated by multiplying the side length by itself: Area = 20 * 20 = 400.
To find the height of the pyramid, we can use the Pythagorean theorem. The diagonal of the square base can be calculated by multiplying the side length by the square root of 2: Diagonal = 20 * √2.
Since the diagonal of the base is directly above the midpoint of the central axis of the cylinder, we have a right-angled triangle formed by the height, half the diagonal of the base and the radius of the cylinder.
The radius of the cylinder is half of its diameter, so r = 10 / 2 = 5.
Using the Pythagorean theorem, we have:
(height)^2 + (r)^2 = (half diagonal of the base)^2
(height)^2 + 5^2 = (20 * √2 / 2)^2
(height)^2 + 25 = (10 * √2)^2
(height)^2 + 25 = 100 * 2
(height)^2 + 25 = 200
(height)^2 = 200 - 25
(height)^2 = 175
height = √175
Now we can substitute the values into the formula for the volume of the pyramid:
V = (1/3) * 400 * √175
V = (400/3) * √175
Volume of a cylinder:
The volume of a cylinder is given by the formula V = π * radius^2 * height.
Since the diameter of the cylinder is 10, the radius is 10 / 2 = 5, and the length is 10.
Substituting the values into the volume formula:
V = π * 5^2 * 10
V = π * 25 * 10
V = 250π
Finally, to find the volume of the space inside the pyramid that is not occupied by the cylinder, we subtract the volume of the cylinder from the volume of the pyramid:
Volume inside the pyramid = (400/3) * √175 - 250π
That's the final answer.