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Classic optimization: wire, window, fence

Problem 3.697 · medium

A closed cylindrical can must hold \( \displaystyle 54 \pi \) cubic inches. What radius \( \displaystyle x \) uses the least material, and what is that surface area?

h = 54*pi/(πx²) from the volume, so S(x) = 2πx² + 2πx·h = 2πx² + 108π/x.
Reviewed
\[ \frac{d}{d x} \left(2 \pi x^{2} + \frac{108 \pi}{x}\right) = \frac{4 \pi \left(x^{3} - 27\right)}{x^{2}} \]
Differentiate.✓ Proved
\[ \left. \frac{4 \pi \left(x^{3} - 27\right)}{x^{2}} \right|_{\substack{ x=3 }} = 0 \]
The only critical point in the domain is x = 3.✓ Proved
\[ \left. \frac{d^{2}}{d x^{2}} \left(2 \pi x^{2} + \frac{108 \pi}{x}\right) \right|_{\substack{ x=3 }} = 12 \pi \]
The second derivative is positive: a minimum.✓ Proved
\[ \left. 2 \pi x^{2} + \frac{108 \pi}{x} \right|_{\substack{ x=3 }} = 54 \pi \]
The surface area there.✓ Proved

Answer \( x = 3,\ \text{surface area} = 54 \pi \approx 169.6460 \)

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 — 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.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — 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.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 4 contains a syntax error in the derivative command: `Derivative(..., (x, 2))` is not standard notation for a second derivative and likely fails or returns an incorrect object in most CAS systems. Additionally, the text claims the second derivative is positive based on this flawed line, but does not explicitly state the value of the second derivative at x=3 (which is 4*pi), relying on a malformed command.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly sets up the surface area function, identifies the critical point, verifies it is a minimum using the second derivative test, and calculates the correct surface area.
gpt-oss:20b: fail (error) 2026-10-14 — The surface area expression is incorrect: 2πx·h = 2πx·(54/x) = 108π, not 108π/x. This propagates through the derivative, critical point, and final surface area calculation.

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.