Home›Calculus 1›Optimization›Problem 3.87

Optimization

Problem 3.87 · medium

A farmer has 200 m of fencing to enclose a rectangular pen. What dimensions maximize the area, and what is the maximum area?

Let the sides be x and y. Then 2x + 2y = 200, so y = 100 − x, and the area is A(x) = x(100 − x) for 0 < x < 100.
Reviewed
\[ \frac{d}{d x} x \left(100.0 - x\right) = 100.0 - 2 x \]
A'(x).✓ Proved
\[ \left. 100.0 - 2 x \right|_{\substack{ x=50.0 }} = 0 \]
A'(x) = 0 at x = 50.0000000000000.✓ Proved
\[ \frac{d^{2}}{d x^{2}} x \left(100.0 - x\right) = -2 \]
A'' < 0, so this is a maximum.✓ Proved
\[ \left. x \left(100.0 - x\right) \right|_{\substack{ x=50.0 }} = 2500.0 \]
A square 50.0000000000000 m by 50.0000000000000 m.✓ Proved

Answer \( 2500.0 \)

✓ 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 sets up the optimization problem, identifies the critical point, verifies it is a maximum using the second derivative test, and calculates the correct maximum area.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly sets up 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 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly sets up 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 sets up 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.