Hydrostatic force
Problem 5.224 · easy
A vertical plate shaped like a triangle with a 6-ft horizontal top edge and its vertex 4 ft below that edge is submerged in water with its top 2 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) = 9 - 3*y/2 and pressure 312/5·y.
- \[ \int\limits_{2}^{6} \frac{312 y \left(9 - \frac{3 y}{2}\right)}{5}\, dy = 2496 \]Force = ∫ (pressure)(width) dy over the plate.✓ Proved
Answer \( 2496 \approx 2496\ \text{lb} \)
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) is incorrect. At y=2 (top of plate), the width should be 6, but the formula gives 9 - 3 = 6 (correct). At y=6 (bottom of pla
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The width function w(y) is incorrect. At y=2 (top of plate), the width should be 6, but the formula gives 9 - 3 = 6 (correct). At y=6 (bottom of plagpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The width function w(y) = 9 - 3y/2 is incorrect. At the top of the plate (y=2), the width should be 6, but the formula gives 7.5. The correct width, derived from similar triangles with vertex at y=6, is w(y) = 3(y-2). Additionally, the integral setup omits the height of the strip (dy is implied but the pressure term needs to be multiplied by the strip area, which is width * dy, so the integrand is correct in form but the width function is wrong).gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.