Classic optimization: wire, window, fence
Problem 3.693 · medium
An open box is made from a \( \displaystyle 24 \times 24 \) sheet by cutting equal squares of side \( \displaystyle x \) from the corners and folding up the sides. What \( \displaystyle x \) gives the largest volume, and what is it?
- V(x) = x(24 − 2x)², for 0 < x < 12.Reviewed
- \[ \frac{d}{d x} x \left(24 - 2 x\right)^{2} = \left(x - 4\right) \left(12 x - 144\right) \]Differentiate.✓ Proved
- \[ \left. 12 \left(x - 12\right) \left(x - 4\right) \right|_{\substack{ x=4 }} = 0 \]The only critical point in the domain is x = 4.✓ Proved
- \[ \left. \frac{d^{2}}{d x^{2}} x \left(24 - 2 x\right)^{2} \right|_{\substack{ x=4 }} = -96 \]The second derivative is negative: a maximum.✓ Proved
- \[ \left. x \left(24 - 2 x\right)^{2} \right|_{\substack{ x=4 }} = 1024 \]The volume there.✓ Proved
Answer \( x = 4,\ \text{volume} = 1024 \)
✓ Nihil obstat Lines: 4 proved, 1 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the objective sampled at 200,000 points across the domain |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly models the volume, identifies the critical point, verifies it is a maximum using the second derivative test, and calculates the correct maximum volume.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly models the volume, identifies the critical point, verifies it is a maximum using the second derivative test, and calculates the correct volume.gpt-oss:20b: pass 2026-10-14
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_classic, checked 2026-10-18 with SymPy 1.14.0.