Answer: et's break down the questions one by one:
2. Faces, vertices, and edges for the cubes and octahedrons
A cube has:
6 faces (top, bottom, left, right, front, and back)
8 vertices (corners)
12 edges (lines connecting the vertices)
An octahedron has:
8 faces (triangular)
6 vertices (corners)
12 edges (lines connecting the vertices)
Euler's Formula states that for any polyhedron, the number of vertices (V), edges (E), and faces (F) are related by the equation:
V - E + F = 2
Let's verify this for both shapes:
Cube: V = 8, E = 12, F = 6 8 - 12 + 6 = 2 (satisfies Euler's Formula)
Octahedron: V = 6, E = 12, F = 8 6 - 12 + 8 = 2 (satisfies Euler's Formula)
3. Percent of empty space in the game container
To find the percent of empty space, we need to know the volume of the game pieces and the volume of the game container. Unfortunately, the problem doesn't provide these values. If you could provide more information about the game pieces and container, I'd be happy to help you with this question.
4. Dimensions of the shipping boxes
Since the manufacturer wants to ship the game in boxes of 12, we need to find a rectangular prism that can hold 12 game pieces. The dimensions of the box will depend on the size and shape of the game pieces.
Let's assume the cube has a side length of x and the octahedron has a side length of y (since it's a triangular face, we can use the length of one side as a reference).
A possible arrangement for the 12 game pieces in a rectangular prism could be:
3 layers of 4 cubes each (3x × 4x × x)
2 layers of 6 octahedrons each (2y × 3y × y)
To find the dimensions of the box, we need to consider the size and shape of the game pieces. If you could provide more information about the game pieces, I'd be happy to help you with this question.
Step-by-step explanation:
