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Optimization

Problem 3.84 · medium

A rectangular field along a straight river needs no fence on the river side. With 240 m of fencing, what is the largest area that can be enclosed?

Let x be the two sides perpendicular to the river; the side parallel is 240 − 2x. Area A(x) = x(240 − 2x), 0 < x < 120.
Reviewed
\[ \frac{d}{d x} x \left(240 - 2 x\right) = 240 - 4 x \]
A'(x).✓ Proved
\[ \left. 240 - 4 x \right|_{\substack{ x=60 }} = 0 \]
A'(x) = 0 at x = 60.✓ Proved
\[ \frac{d^{2}}{d x^{2}} x \left(240 - 2 x\right) = -4 \]
A'' < 0: a maximum.✓ Proved
\[ \left. x \left(240 - 2 x\right) \right|_{\substack{ x=60 }} = 7200 \]
60 m by 120 m.✓ Proved

Answer \( 7200 \)

✓ 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 200,001 points of its interval tops out at the same value

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly models the problem, identifies the critical point, verifies it is a maximum using the second derivative test, and calculates the correct area.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly models the problem, identifies the critical point, verifies it is a maximum using the second derivative test, and calculates the correct area.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly models the optimization problem, identifies the critical point, verifies it is a maximum using the second derivative test, and calculates the correct maximum area.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly models the optimization problem, identifies the critical point, verifies it is a maximum using the second derivative test, and calculates the correct maximum area.
gpt-oss:20b: pass 2026-09-26

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, checked 2026-09-28 with SymPy 1.14.0.