Home›Calculus 1›Classic optimization: wire, window, fence›Problem 3.694

Classic optimization: wire, window, fence

Problem 3.694 · medium

A rectangular pen of area \( \displaystyle 5400 \) m² is split in two by a fence parallel to two of its sides (\( \displaystyle x \) is the length of those sides). What \( \displaystyle x \) uses the least fencing, and how much is that?

With sides x and 5400/x, the fence is three x-sides and two others: F(x) = 3x + 2·5400/x.
Reviewed
\[ \frac{d}{d x} \left(3 x + \frac{10800}{x}\right) = 3 - \frac{10800}{x^{2}} \]
Differentiate.✓ Proved
\[ \left. 3 - \frac{10800}{x^{2}} \right|_{\substack{ x=60 }} = 0 \]
The only critical point in the domain is x = 60.✓ Proved
\[ \left. \frac{d^{2}}{d x^{2}} \left(3 x + \frac{10800}{x}\right) \right|_{\substack{ x=60 }} = \frac{1}{10} \]
The second derivative is positive: a minimum.✓ Proved
\[ \left. 3 x + \frac{10800}{x} \right|_{\substack{ x=60 }} = 360 \]
The fencing there.✓ Proved

Answer \( x = 60,\ \text{fencing} = 360 \)

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 models the fencing cost function and applies standard calculus techniques to find the minimum. The setup, differentiation, and verification steps are all sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly models the fencing cost function and applies standard calculus techniques to find the minimum. The setup, differentiation, and verification steps are all sound.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly models the fencing problem, identifies the critical point, verifies it is a minimum using the second derivative test, and calculates the correct minimum fencing amount.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly models the fencing problem, identifies the critical point, verifies it is a minimum using the second derivative test, and calculates the correct minimum fencing amount.
gpt-oss:20b: fail (error) 2026-10-14 — The setup incorrectly counts the fencing lengths. Since the fence is parallel to the sides of length x, the fence itself has length 5400/x, not x. Thus the total fencing should be 2x + 3·(5400/x), not 3x + 2·(5400/x).

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.