Classic optimization: wire, window, fence
Problem 3.677 · medium
An open box is made from a \( \displaystyle 30 \times 30 \) 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(30 − 2x)², for 0 < x < 15.Reviewed
- \[ \frac{d}{d x} x \left(30 - 2 x\right)^{2} = \left(x - 5\right) \left(12 x - 180\right) \]Differentiate.✓ Proved
- \[ \left. 12 \left(x - 15\right) \left(x - 5\right) \right|_{\substack{ x=5 }} = 0 \]The only critical point in the domain is x = 5.✓ Proved
- \[ \left. \frac{d^{2}}{d x^{2}} x \left(30 - 2 x\right)^{2} \right|_{\substack{ x=5 }} = -120 \]The second derivative is negative: a maximum.✓ Proved
- \[ \left. x \left(30 - 2 x\right)^{2} \right|_{\substack{ x=5 }} = 2000 \]The volume there.✓ Proved
Answer \( x = 5,\ \text{volume} = 2000 \)
✓ 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 — 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.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — 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 2026-10-17qwen3.6:27b-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. The domain restriction is appropriate.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly models the volume, identifies the critical point within the valid domain, and verifies it is a maximum using the second derivative test. The final calculation is correct.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 2026-10-17 with SymPy 1.14.0.