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Classic optimization: wire, window, fence

Problem 3.667 · medium

A Norman window is a rectangle topped by a semicircle. If its perimeter is \( \displaystyle 8 \) ft, what radius \( \displaystyle x \) of the semicircle lets in the most light (the largest area), and what is that area?

Width 2x, height h = (8 − 2x − πx)/2 from the perimeter; A(x) = 2x·h + πx²/2.
Reviewed
\[ \frac{d}{d x} \left(\frac{\pi x^{2}}{2} + 2 x \left(- \frac{\pi x}{2} - x + 4\right)\right) = - 4 x - \pi x + 8 \]
Differentiate.✓ Proved
\[ \left. - 4 x - \pi x + 8 \right|_{\substack{ x=\frac{8}{\pi + 4} }} = 0 \]
The only critical point in the domain is x = 8/(pi + 4).✓ Proved
\[ \left. \frac{d^{2}}{d x^{2}} \left(\frac{\pi x^{2}}{2} + 2 x \left(\frac{- \pi x}{2} - x + 4\right)\right) \right|_{\substack{ x=\frac{8}{\pi + 4} }} = -4 - \pi \]
The second derivative is negative: a maximum.✓ Proved
\[ \left. \frac{\pi x^{2}}{2} + 2 x \left(\frac{- \pi x}{2} - x + 4\right) \right|_{\substack{ x=\frac{8}{\pi + 4} }} = \frac{32}{\pi + 4} \]
The area there.✓ Proved

Answer \( x = \frac{8}{\pi + 4},\ \text{area} = \frac{32}{\pi + 4} \approx 4.4808 \)

✓ 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: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly models the geometry, derives the area function, and identifies the maximum using standard calculus techniques. The algebraic steps are verified and the final answer is correct.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly models the perimeter and area, derives the critical point, verifies the maximum via the second derivative, and calculates the final area accurately.
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.