Classic optimization: wire, window, fence
Problem 3.673 · medium
A closed cylindrical can must hold \( \displaystyle 16 \pi \) cubic inches. What radius \( \displaystyle x \) uses the least material, and what is that surface area?
- h = 16*pi/(πx²) from the volume, so S(x) = 2πx² + 2πx·h = 2πx² + 32π/x.Reviewed
- \[ \frac{d}{d x} \left(2 \pi x^{2} + \frac{32 \pi}{x}\right) = \frac{4 \pi \left(x^{3} - 8\right)}{x^{2}} \]Differentiate.✓ Proved
- \[ \left. \frac{4 \pi \left(x^{3} - 8\right)}{x^{2}} \right|_{\substack{ x=2 }} = 0 \]The only critical point in the domain is x = 2.✓ Proved
- \[ \left. \frac{d^{2}}{d x^{2}} \left(2 \pi x^{2} + \frac{32 \pi}{x}\right) \right|_{\substack{ x=2 }} = 12 \pi \]The second derivative is positive: a minimum.✓ Proved
- \[ \left. 2 \pi x^{2} + \frac{32 \pi}{x} \right|_{\substack{ x=2 }} = 24 \pi \]The surface area there.✓ Proved
Answer \( x = 2,\ \text{surface area} = 24 \pi \approx 75.3982 \)
✓ 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 derives the surface area function, identifies the critical point, and verifies it is a minimum using the second derivative test. The final values are correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly derives the surface area function, identifies the critical point, and verifies it is a minimum using the second derivative test. The final values are correct.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly derives the surface area function, identifies the critical point, and verifies it is a minimum using the second derivative test. The final values are correct.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly derives 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
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.