Hydrostatic force
Problem 5.511 · easy
A vertical plate shaped like a triangle with a 4-ft horizontal top edge and its vertex 3 ft below that edge is submerged in water with its top 3 ft below the surface. Find the force of the water on one side. (Water weighs \( \displaystyle \frac{312}{5}\ \text{lb}/\text{ft}^3 \).)
- Let y be depth below the surface. A horizontal strip at depth y has width w(y) = 8 - 4*y/3 and pressure 312/5·y.
- \[ \int\limits_{3}^{6} \frac{312 y \left(8 - \frac{4 y}{3}\right)}{5}\, dy = \frac{7488}{5} \]Force = ∫ (pressure)(width) dy over the plate.✓ Proved
Answer \( \frac{7488}{5} \approx 1497.6\ \text{lb} \)
Lines: 1 proved, 1 not checked. 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ 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 | numerical quadrature |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The width function w(y) = 8 - 4y/3 is incorrect; at the top of the plate (y=3), the width is 4, but the formula gives 4. At the bottom (y=6), the wi
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The width function w(y) = 8 - 4y/3 is incorrect; at the top of the plate (y=3), the width is 4, but the formula gives 4. At the bottom (y=6), the wiqwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The width function w(y) = 8 - 4y/3 is incorrect; at y=3, it gives 4 (correct), but at y=6, it gives 0 (correct), implying the triangle widens as itgpt-oss:20b: pass 2026-10-12
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/hydrostatic_force, checked 2026-10-12 with SymPy 1.14.0.