2 planes parallel to the base of a pyramid cut the pyramid at 1/4 and 2/3 of the height as measured from the base up. If the volume of the pyramid is 100 cubic inches, what is the volume of the middle piece to the nearest tenth?
Volumes of pyramids above the cutting planes are proportional to the cube of the height.
So the volume above the cutting plane at h/4 is
V1=(3/4)^3*100=2700/64 in³
Volume above the 2h/3 cut is
V2=(1/3)^3*100=100/27 in³
The volume between two cuts are therefore
= V1-V2
=100(27/64-1/27)
=16625/432
=38.48 in³ (approx.)
Thank you very much!
You're welcome!
To solve this problem, we need to break it down step by step. Let's start by understanding the given information.
We have a pyramid and two planes that are parallel to the base of the pyramid. These planes cut the pyramid at specific points along its height. The first plane cuts it at 1/4 of the height, while the second plane cuts it at 2/3 of the height.
The volume of the entire pyramid is given as 100 cubic inches. We need to find the volume of the middle piece that is between the two planes.
To find the volume of the middle piece, we first need to determine the heights at which the two planes intersect the pyramid.
Let's assume the height of the pyramid as h.
First Plane: Since it cuts the pyramid at 1/4 of the height, the height at which it intersects is 1/4 * h.
Second Plane: Since it cuts the pyramid at 2/3 of the height, the height at which it intersects is 2/3 * h.
Now, we need to find the heights at which the two planes intersect the base of the pyramid.
Height of First Plane from Base: 1/4 * h
Height of Second Plane from Base: 2/3 * h
Next, we need to find the volumes of two sections formed by the planes and the base of the pyramid.
Volume of First Section: (1/3) * base area * height of the first section
Volume of Second Section: (1/3) * base area * height of the second section
Since the two planes are parallel to the base, they cut out similar triangles on the pyramid's base. So, the base area will remain the same for both sections.
Now, let's substitute the given values:
Volume of First Section: (1/3) * base area * (1/4 * h)
Volume of Second Section: (1/3) * base area * (2/3 * h)
Since we know that the volume of the entire pyramid is 100 cubic inches, the sum of the volumes of both sections should equal 100 cubic inches.
(1/3) * base area * (1/4 * h) + (1/3) * base area * (2/3 * h) = 100
Factoring out the base area, we get:
(1/12 * h) + (2/9 * h) = 100
Now, we can simplify the equation:
(3/36 * h) + (8/36 * h) = 100
(11/36 * h) = 100
Solving for h:
h = (100 * 36) / 11
h ≈ 327.273 cubic inches
Now that we have the height of the pyramid, we can calculate the height of the middle piece.
Height of Middle Piece: (2/3 * h) - (1/4 * h)
Substituting the value of h:
Height of Middle Piece: (2/3 * 327.273) - (1/4 * 327.273)
Height of Middle Piece ≈ 218.182 cubic inches
Finally, to find the volume of the middle piece, we can multiply the area of the base by the height of the middle piece:
Volume of Middle Piece: base area * Height of Middle Piece
Volume of Middle Piece ≈ base area * 218.182 cubic inches
Since we don't have the dimensions of the base, we cannot calculate the exact volume of the middle piece. However, once we know the base dimensions, we can multiply it by 218.182 to find the volume of the middle piece to the nearest tenth.