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Hydrostatic force

Problem 5.293 · easy

A vertical plate shaped like a triangle with a 5-m horizontal top edge and its vertex 4 m below that edge is submerged in water with its top 3 m below the surface. Find the force of the water on one side. (Water weighs \( \displaystyle 9800\ \text{N}/\text{m}^3 \).)

Let y be depth below the surface. A horizontal strip at depth y has width w(y) = 35/4 - 5*y/4 and pressure 9800·y.
\[ \int\limits_{3}^{7} 9800 y \left(\frac{35}{4} - \frac{5 y}{4}\right)\, dy = \frac{1274000}{3} \]
Force = ∫ (pressure)(width) dy over the plate.✓ Proved

Answer \( \frac{1274000}{3} \approx 424667\ \text{N} \)

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 returned a non-object

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The setup correctly models the geometry and physics: the width function w(y) is linear, matching the triangle's dimensions (width 5 at y=3, width 0 at y=7), and the pressure is correctly given by 9800y. The integration limits and result 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 2026-10-06 with SymPy 1.14.0.