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Hydrostatic force

Problem 5.372 · easy

A vertical plate shaped like a triangle with its vertex at the top and a 4-m horizontal base 3 m below the vertex is submerged in water with its top at 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) = 4*y/3 and pressure 9800·y.
Reviewed
\[ \int\limits_{0}^{3} \frac{39200 y^{2}}{3}\, dy = 117600 \]
Force = ∫ (pressure)(width) dy over the plate.✓ Proved

Answer \( 117600 \approx 117600\ \text{N} \)

✓ 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 models the geometry (width varying linearly with depth) and the physics (pressure proportional to depth). The integration limits and integrand are correct, leading to the right answer.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The setup correctly models the geometry (width varying linearly with depth) and the physics (pressure proportional to depth). The integration limits and integrand are correct, leading to the right answer.
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly models the geometry (width varying linearly with depth) and the physics (pressure proportional to depth). The integration setup and result are correct.
gpt-oss:20b: pass 2026-10-07

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 2026-10-07 with SymPy 1.14.0.