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Classic optimization: wire, window, fence

Problem 3.665 · medium

A closed cylindrical can must hold \( \displaystyle 432 \pi \) cubic inches. What radius \( \displaystyle x \) uses the least material, and what is that surface area?

h = 432*pi/(πx²) from the volume, so S(x) = 2πx² + 2πx·h = 2πx² + 864π/x.
Reviewed
\[ \frac{d}{d x} \left(2 \pi x^{2} + \frac{864 \pi}{x}\right) = \frac{4 \pi \left(x^{3} - 216\right)}{x^{2}} \]
Differentiate.✓ Proved
\[ \left. \frac{4 \pi \left(x^{3} - 216\right)}{x^{2}} \right|_{\substack{ x=6 }} = 0 \]
The only critical point in the domain is x = 6.✓ Proved
\[ \left. \frac{d^{2}}{d x^{2}} \left(2 \pi x^{2} + \frac{864 \pi}{x}\right) \right|_{\substack{ x=6 }} = 12 \pi \]
The second derivative is positive: a minimum.✓ Proved
\[ \left. 2 \pi x^{2} + \frac{864 \pi}{x} \right|_{\substack{ x=6 }} = 216 \pi \]
The surface area there.✓ Proved

Answer \( x = 6,\ \text{surface area} = 216 \pi \approx 678.5840 \)

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 — The solution correctly sets up the surface area function, identifies the critical point, and verifies it is a minimum using the second derivative test. The final calculation is correct.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly derives the surface area function, identifies the critical point, verifies it is a minimum using the second derivative test, and calculates the final surface area. All steps are logically sound and algebraically correct.
gpt-oss:20b: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The second derivative is mis‑calculated: it should be \(16\pi/3\), not \(12\pi\)."}

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.