Classic optimization: wire, window, fence
Problem 3.670 · medium
A Norman window is a rectangle topped by a semicircle. If its perimeter is \( \displaystyle 20 \) 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 = (20 − 2x − πx)/2 from the perimeter; A(x) = 2x·h + πx²/2.Reviewed
- \[ \frac{d}{d x} \left(\frac{\pi x^{2}}{2} + 2 x \left(- \frac{\pi x}{2} - x + 10\right)\right) = - 4 x - \pi x + 20 \]Differentiate.✓ Proved
- \[ \left. - 4 x - \pi x + 20 \right|_{\substack{ x=\frac{20}{\pi + 4} }} = 0 \]The only critical point in the domain is x = 20/(pi + 4).✓ Proved
- \[ \left. \frac{d^{2}}{d x^{2}} \left(\frac{\pi x^{2}}{2} + 2 x \left(\frac{- \pi x}{2} - x + 10\right)\right) \right|_{\substack{ x=\frac{20}{\pi + 4} }} = -4 - \pi \]The second derivative is negative: a maximum.✓ Proved
- \[ \left. \frac{\pi x^{2}}{2} + 2 x \left(\frac{- \pi x}{2} - x + 10\right) \right|_{\substack{ x=\frac{20}{\pi + 4} }} = \frac{200}{\pi + 4} \]The area there.✓ Proved
Answer \( x = \frac{20}{\pi + 4},\ \text{area} = \frac{200}{\pi + 4} \approx 28.0050 \)
✓ 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 models the perimeter and area, identifies the critical point, verifies the maximum via the second derivative, and calculates the final area accurately.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly models the perimeter and area, identifies the critical point, verifies the maximum via the second derivative, and calculates the final area accurately.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly models the perimeter and area, identifies the critical point, verifies the maximum using the second derivative test, and calculates the correct maximum area.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly models the perimeter and area, performs the differentiation and optimization accurately, and arrives at the correct final values.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.