1.

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

Akhaim1 Jan 31, 2024

#2**0 **

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 23s. Since ABCD has volume 18, we can find s using the formula for the volume of a tetrahedron:

31s2⋅23s=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=63s.

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 2s. Therefore, the side length of the square base of EFGH is 22s=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)

BuiIderBoi Feb 6, 2024

#3**0 **

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)

BuiIderBoi Feb 6, 2024