If a pyramid is sliced by a plane parallel to the base of the pyramid and that plane bisects the altitude of the pyramid, what percentage of the volume of the pyramid lies below the slicing plane?
To find the percentage of the volume of the pyramid that lies below the slicing plane, we can use the concept of similar triangles.
Let's denote the original pyramid as P and the sliced pyramid as P'.
When the slicing plane is parallel to the base of the pyramid and bisects the altitude, we know that the two resulting pyramid halves are similar.
Since the slicing plane bisects the altitude, the height of pyramid P' is half of the height of pyramid P.
Now, let's think about the volume of the pyramid. The volume of a pyramid is given by the formula V = (1/3) * (base area) * (height).
Since the base area remains the same for both P and P' (as the slicing plane is parallel to the base), the volume of pyramid P' is half of the volume of pyramid P because its height is half of the height of pyramid P.
Therefore, the percentage of the volume of the pyramid lying below the slicing plane can be calculated as:
( Volume of pyramid P' / Volume of pyramid P ) * 100
= ( 1/2 ) * 100
= 50%
Therefore, 50% of the volume of the pyramid lies below the slicing plane.
To solve this problem, we need to understand the concept of similar triangles.
Let's consider a pyramid with a base area B and height h. When the pyramid is sliced by a plane parallel to the base and bisecting the altitude, it forms two smaller pyramids.
Since the slicing plane bisects the altitude, each of the resulting pyramids will have half the height of the original pyramid, which is h/2. Moreover, the base area of each smaller pyramid will be half of the base area of the original pyramid, which is B/2.
Now, let's look at the volume of the pyramid. The volume of a pyramid is calculated by the formula V = (1/3) * B * h, where B represents the base area and h represents the height.
For the original pyramid, its volume is V1 = (1/3) * B * h.
For each smaller pyramid, the volume is V2 = (1/3) * (B/2) * (h/2) = (1/12) * B * h.
Therefore, the total volume of both smaller pyramids is V2 + V2 = (2/12) * B * h = (1/6) * B * h.
To find the percentage of the volume of the pyramid that lies below the slicing plane, we need to calculate the ratio of the volume below the slicing plane to the total volume of the pyramid.
The volume below the slicing plane is V2 + V2 = (1/6) * B * h, and the total volume of the pyramid is V1 = (1/3) * B * h.
So, the percentage of the volume that lies below the slicing plane is ((1/6) * B * h) / ((1/3) * B * h) = (1/6) / (1/3) = 1/2 = 50%.
Therefore, 50% of the volume of the pyramid lies below the slicing plane.
the volumes of similar shapes is proportional to the cubes of the corresponding heights.
So the small pyramid cut off at the top is to the whole pyramid in the ratio of 1 : 8
So the bottom part is 7/8 of the whole or 87.5%