Classic optimization: wire, window, fence
Problem 3.668 · medium
A wire \( \displaystyle 10 \) m long is cut in two. One piece (length \( \displaystyle x \)) is bent into a square, the other into a circle. What \( \displaystyle x \) makes the total area as small as possible, and what is that area?
- Square side x/4; circle circumference 10 − x, radius (10 − x)/(2π). A(x) = (x/4)² + (10 − x)²/(4π).Reviewed
- \[ \frac{d}{d x} \left(\frac{x^{2}}{16} + \frac{\left(10 - x\right)^{2}}{4 \pi}\right) = \frac{\pi x + 4 x - 40}{8 \pi} \]Differentiate.✓ Proved
- \[ \left. \frac{\pi x + 4 x - 40}{8 \pi} \right|_{\substack{ x=\frac{40}{\pi + 4} }} = 0 \]The only critical point in the domain is x = 40/(pi + 4).✓ Proved
- \[ \left. \frac{d^{2}}{d x^{2}} \left(\frac{x^{2}}{16} + \frac{\left(10 - x\right)^{2}}{4 \pi}\right) \right|_{\substack{ x=\frac{40}{\pi + 4} }} = \frac{\pi + 4}{8 \pi} \]The second derivative is positive: a minimum.✓ Proved
- \[ \left. \frac{x^{2}}{16} + \frac{\left(10 - x\right)^{2}}{4 \pi} \right|_{\substack{ x=\frac{40}{\pi + 4} }} = \frac{25}{\pi + 4} \]The total area there.✓ Proved
Answer \( x = \frac{40}{\pi + 4},\ \text{total area} = \frac{25}{\pi + 4} \approx 3.5006 \)
✓ 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 2026-10-17gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly models the problem, identifies the critical point, verifies it is a minimum using the second derivative test, and calculates the correct minimum area.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.