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Hydrostatic force

Problem 5.231 · easy

A vertical plate shaped like a rectangle 4 ft wide and 4 ft tall 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) = 4 and pressure 312/5·y.
Reviewed
\[ \int\limits_{0}^{4} \frac{1248 y}{5}\, dy = \frac{9984}{5} \]
Force = ∫ (pressure)(width) dy over the plate.✓ Proved

Answer \( \frac{9984}{5} \approx 1996.8\ \text{lb} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

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 identifies the pressure as a function of depth and the width of the plate. The integration limits and resulting calculation are correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The setup correctly identifies the pressure as a function of depth and the width of the plate. The integration limits and resulting calculation are correct.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The setup correctly identifies the pressure as a function of depth and the width of the plate. The integration bounds and the resulting calculation are correct.
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.