Optimization
Problem 3.88 · medium
Equal squares are cut from the corners of a 6 in by 6 in sheet of cardboard and the sides folded up to make an open box. What cut maximizes the volume, and what is the volume?
- Cutting squares of side x leaves a base 6 − 2x on a side and height x: V(x) = x(6 − 2x)², 0 < x < 3.Reviewed
- \[ \frac{d}{d x} x \left(6 - 2 x\right)^{2} = \left(x - 1\right) \left(12 x - 36\right) \]V'(x), factored.✓ Proved
- \[ \left. x \left(8 x - 24\right) + \left(6 - 2 x\right)^{2} \right|_{\substack{ x=1 }} = 0 \]V'(x) = 0 at x = 1 (the other root, x = 3, gives no box).✓ Proved
- \[ \left. x \left(6 - 2 x\right)^{2} \right|_{\substack{ x=1 }} = 16 \]The maximum volume.✓ Proved
- V is 0 at both ends of the interval and positive in between, so the interior critical point is the maximum.Reviewed
Answer \( 16 \)
✓ Nihil obstat Lines: 3 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the objective sampled at 200,001 points of its interval tops out at the same value |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly models the problem, identifies the critical point, and justifies the maximum using the boundary conditions. The final answer matches the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly models the problem, identifies the critical point, and justifies the maximum using the boundary conditions. The final answer matches the stated answer.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly models the problem, identifies the critical point, and justifies it as a maximum using the boundary behavior. The final answer matches the stated answer.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly models the problem, identifies the critical point, and justifies the maximum using the boundary behavior of the volume function.gpt-oss:20b: pass 2026-09-26
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/optimization, checked 2026-09-28 with SymPy 1.14.0.