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Classic optimization: wire, window, fence

Problem 3.669 · medium

A wire \( \displaystyle 20 \) 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 20 − x, radius (20 − x)/(2π). A(x) = (x/4)² + (20 − x)²/(4π).
Reviewed
\[ \frac{d}{d x} \left(\frac{x^{2}}{16} + \frac{\left(20 - x\right)^{2}}{4 \pi}\right) = \frac{\pi x + 4 x - 80}{8 \pi} \]
Differentiate.✓ Proved
\[ \left. \frac{\pi x + 4 x - 80}{8 \pi} \right|_{\substack{ x=\frac{80}{\pi + 4} }} = 0 \]
The only critical point in the domain is x = 80/(pi + 4).✓ Proved
\[ \left. \frac{d^{2}}{d x^{2}} \left(\frac{x^{2}}{16} + \frac{\left(20 - x\right)^{2}}{4 \pi}\right) \right|_{\substack{ x=\frac{80}{\pi + 4} }} = \frac{\pi + 4}{8 \pi} \]
The second derivative is positive: a minimum.✓ Proved
\[ \left. \frac{x^{2}}{16} + \frac{\left(20 - x\right)^{2}}{4 \pi} \right|_{\substack{ x=\frac{80}{\pi + 4} }} = \frac{100}{\pi + 4} \]
The total area there.✓ Proved

Answer \( x = \frac{80}{\pi + 4},\ \text{total area} = \frac{100}{\pi + 4} \approx 14.0025 \)

Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. 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: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer returned a non-object
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.