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Hydrostatic force

Problem 5.508 · easy

A vertical plate shaped like a triangle with a 2-ft horizontal top edge and its vertex 4 ft below that edge is submerged in water with its top at 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) = 2 - y/2 and pressure 312/5·y.
\[ \int\limits_{0}^{4} \frac{312 y \left(2 - \frac{y}{2}\right)}{5}\, dy = \frac{1664}{5} \]
Force = ∫ (pressure)(width) dy over the plate.✓ Proved

Answer \( \frac{1664}{5} \approx 332.8\ \text{lb} \)

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: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The width function w(y) = 2 - y/2 is incorrect. At y=0, width is 2 (correct), but at y=4, width is 0 (correct), implying a linear decrease. However,

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The width function w(y) = 2 - y/2 is incorrect. At y=0, width is 2 (correct), but at y=4, width is 0 (correct), implying a linear decrease. However,
gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: pass 2026-10-12 — The setup correctly models the geometry (width varying linearly from 2 to 0) and the physics (pressure proportional to depth). The integration limits and integrand are correct.

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.