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Hydrostatic force

Problem 5.442 · easy

A vertical plate shaped like a triangle with a 3-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) = 9/2 - 3*y/4 and pressure 312/5·y.
Reviewed
\[ \int\limits_{2}^{6} \frac{312 y \left(\frac{9}{2} - \frac{3 y}{4}\right)}{5}\, dy = 1248 \]
Force = ∫ (pressure)(width) dy over the plate.✓ Proved

Answer \( 1248 \approx 1248\ \text{lb} \)

Lines: 1 proved, 1 reviewed. 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	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
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: pass — The setup correctly models the geometry (width as a function of depth) and the physics (hydrostatic pressure). The integration bounds and result are correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The setup correctly models the geometry (width as a function of depth) and the physics (hydrostatic pressure). The integration bounds and result are correct.
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The width function w(y) is incorrect. At y=2 (top of plate), the width should be 3, but the formula gives 9/2 - 3(2)/4 = 1.5. The correct width is w(y) = 3(y-6)/4 or similar, derived from similar triangles where width is 0 at y=6 and 3 at y=2.
gpt-oss:20b: pass 2026-10-09

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 2026-10-09 with SymPy 1.14.0.