Hydrostatic force
Problem 5.228 · 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 2 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) = 4*y/3 - 8/3 and pressure 9800·y.
- \[ \int\limits_{2}^{5} 9800 y \left(\frac{4 y}{3} - \frac{8}{3}\right)\, dy = 235200 \]Force = ∫ (pressure)(width) dy over the plate.✓ Proved
Answer \( 235200 \approx 235200\ \text{N} \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The width function w(y) is incorrect. At the top of the plate (y=2), the width should be 0, but the given formula yields -4/3. The correct width, based on similar triangles with the vertex at y=-1, is w(y) = 4(y+1)/3.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.