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Hydrostatic force

Problem 5.225 · easy

A vertical plate shaped like a rectangle 4 ft wide and 3 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
\[ \int\limits_{0}^{3} \frac{1248 y}{5}\, dy = \frac{5616}{5} \]
Force = ∫ (pressure)(width) dy over the plate.✓ Proved

Answer \( \frac{5616}{5} \approx 1123.2\ \text{lb} \)

✓ 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 width as constant and pressure as linear with depth. The integration limits and integrand are correct, leading to the right answer.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The setup correctly identifies the width as constant and pressure as linear with depth. The integration limits and integrand are correct, leading to the right answer.
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 limits and 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.