Hydrostatic force
Problem 5.368 · easy
A vertical plate shaped like a triangle with its vertex at the top and a 2-m horizontal base 1 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) = 2*y - 4 and pressure 9800·y.
- \[ \int\limits_{2}^{3} 9800 y \left(2 y - 4\right)\, dy = \frac{78400}{3} \]Force = ∫ (pressure)(width) dy over the plate.✓ Proved
Answer \( \frac{78400}{3} \approx 26133.3\ \text{N} \)
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 response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The width function w(y) = 2y - 4 is incorrect; at the top of the plate (y=2), the width should be 0, but 2(2)-4=0 is correct, however, at the bottom
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The width function w(y) = 2y - 4 is incorrect; at the top of the plate (y=2), the width should be 0, but 2(2)-4=0 is correct, however, at the bottomqwen3.6:27b-mlx: fail (error) 2026-10-07 — The width function w(y) = 2y - 4 is incorrect; at y=2 (top of plate), width is 0, but the problem states the vertex is at the top, implying width 0 at the top. However, the base is 2m wide at y=3 (1m below vertex). Using similar triangles, width should be proportional to distance from vertex. If y is depth, vertex is at y=2. Distance from vertex is y-2. Width w(y) = 2 * (y-2) / 1 = 2y - 4. Wait, let's re-read carefully. Vertex at top. Base 2m horizontal, 1m below vertex. Top of plate is 2m below surface. So vertex is at y=2. Base is at y=3. Width at y=2 is 0. Width at y=3 is 2. The formula w(y) = 2(y-2) = 2y-4 is correct for the geometry. Let's check the integral. Integral from 2 to 3 of 9800 * y * (2y-4) dy. 9800 * Integral(2y^2 - 4y) dy from 2 to 3. Antiderivative: 2/3 y^3 - 2y^2. Evaluated at 3: 2/3(27) - 2(9) = 18 - 18 = 0. Evaluated at 2: 2/3(8) - 2(4) = 16/3 - 8 = -8/3. Result: 0 - (-8/3) = 8/3. Total Force = 9800 * 8/3 = 78400/3. The math is actually correct. Why did I think it was wrong? Let me re-read the prompt's implied trap. 'Vertex at the top'. 'Base 2m horizontal 1m below vertex'. 'Top 2m below surface'. So y=2 is vertex. y=3 is base. Width at y is linear. w(2)=0, w(3)=2. Slope = 2/1 = 2. w(y) = 2(y-2) = 2y-4. This is correct. Pressure is 9800y. Integral is correct. The solution is sound.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.