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# 3D Geometry

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1.

2.There was another person who asked the first problem but the person who answered was wrong

Jan 31, 2024

#2
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Problem 1

Since ABCD is a regular tetrahedron, each face is an equilateral triangle with side length s. It follows that the altitude of each face is 23​​s. Since ABCD has volume 18, we can find s using the formula for the volume of a tetrahedron:

31​s2⋅23​​s=18⇒s=6

Now, let's focus on the smaller pyramid EFGH. Each of its faces is a triangle similar to the faces of ABCD (with side length 3s​). So, the altitude of each face is also 23​​⋅3s​=63​​s.

Furthermore, the base of EFGH is a square formed by connecting the midpoints of the edges of ABCD. Since ABCD is regular, its diagonals bisect each other at right angles, and their length is 2​s. Therefore, the side length of the square base of EFGH is 22​s​=32​.

Finally, the volume of EFGH can be calculated using the formula for the volume of a pyramid:

1/3*(3*sqrt(2))^3*sqrt(3)/6*s = 3*sqrt(6)/2 = 9*sqrt(6)

Feb 6, 2024
#3
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Problem 2

Let's analyze what happens when Donatello slices off a pyramid from each corner of the cube:

Volume of each pyramid: Each pyramid has a square base with side length equal to half the edge length of the cube, which is 3 in this case. The height of each pyramid is also half the diagonal of the cube's face.

Since the cube is a regular hexahedron, the diagonal of each face is equal to 2​ times the side length, which is 62​ in this case. Therefore, the height of each pyramid is 32​/2. Using the formula for the volume of a pyramid, we find the volume of each removed pyramid:

Volume of each pyramid = (1/3) * base area * height = (1/3) * (3)^2 * (3 * sqrt(2)/2) = 9 * sqrt(2)

Remaining volume: Since there are 8 corner points in a cube, Donatello removes 8 such pyramids. The total volume removed is 8 times the volume of one pyramid:

Total volume removed = 8 * 9 * sqrt(2) = 72 * sqrt(2)

Volume of remaining polyhedron: The original cube had a volume of 6^3 = 216. Subtracting the volume removed due to the pyramids, we find the volume of the remaining polyhedron:

Volume of remaining polyhedron = 216 - 72 * sqrt(2)

Feb 6, 2024