Answer:
Explanation:
When a cube is cut parallel to one face with 7 cuts such that all resulting pieces are identical, we can visualize the cube as a stack of identical layers.
To find the maximum number of identical pieces that can be obtained by making 9 more cuts in any direction, we need to consider the number of sections created by each cut.
Let's break it down step by step:
1. The initial 7 cuts create 8 identical pieces, forming the stack of layers.
2. Now, we need to consider the additional 9 cuts. Each cut will divide each layer into multiple sections.
3. To maximize the number of identical pieces, we want each cut to intersect with as many previous cuts as possible.
4. If we consider each layer individually, the maximum number of sections created by a single cut is equal to the number of previous cuts it intersects with plus 1.
5. Therefore, the first additional cut will create 1 + 7 = 8 sections (intersecting with all 7 previous cuts plus the initial cut).
6. The second additional cut will create 2 + 7 = 9 sections (intersecting with all 7 previous cuts plus the initial cut).
7. Continuing this pattern, the ninth additional cut will create 9 + 7 = 16 sections (intersecting with all 7 previous cuts plus the initial cut).
8. Finally, we sum up the number of sections created by each cut to find the maximum number of identical pieces. In this case, the maximum number of identical pieces that can be obtained by making 9 more cuts is:
8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 = 108
Therefore, the maximum number of identical pieces that can be obtained by making 9 more cuts in any direction is 108.
I hope this explanation helps! If you have any further questions, feel free to ask.