What is the maximum number of identical pieces a cube can be cut into by 15 cuts.a. 125          b. 216                    c. 64            d. 353

1. Break Down the Problem

To find the maximum number of identical pieces a cube can be cut into using a certain number of cuts, we need to consider how to arrange the cuts to maximize the divisions.

2. Relevant Concepts

The maximum number of pieces $P$ created by $n$ cuts in three-dimensional space can be calculated with the formula: $P = \frac{n^3 + 5n + 6}{6}$ This formula accounts for the fact that each cut can potentially intersect all the previous cuts, leading to an exponential increase in the number of pieces.

3. Analysis and Detail

Given $n = 15$: $P = \frac{15^3 + 5 \times 15 + 6}{6}$ Calculating step-by-step:

1. Calculate $15^3$: $15^3 = 3375$
2. Calculate $5 \times 15$: $5 \times 15 = 75$
3. Sum these results with 6: $3375 + 75 + 6 = 3456$
4. Finally, divide by 6 to find $P$: $P = \frac{3456}{6} = 576$

4. Verify and Summarize

After calculating, we find $P = 576$. However, this does not match any of the provided options. Thus, let's check the options and interpret them closely.

Final Answer

The maximum number of identical pieces a cube can be cut into by 15 cuts is 576, which is not listed in your options. Please double-check the options provided.