Hydrostatic force
Problem 5.296 · easy
A vertical plate shaped like a triangle with a 4-m horizontal top edge and its vertex 1 m below that edge 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) = 12 - 4*y and pressure 9800·y.
- \[ \int\limits_{2}^{3} 9800 y \left(12 - 4 y\right)\, dy = \frac{137200}{3} \]Force = ∫ (pressure)(width) dy over the plate.✓ Proved
Answer \( \frac{137200}{3} \approx 45733.3\ \text{N} \)
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) = 12 - 4y is incorrect. At y=2, the width should be 4, but the formula gives 4. At y=3, the width should be 0, but the formu
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The width function w(y) = 12 - 4y is incorrect. At y=2, the width should be 4, but the formula gives 4. At y=3, the width should be 0, but the formuqwen3.6:27b-mlx: fail (error) 2026-10-06 — The width function w(y) = 12 - 4y is incorrect. At the top of the plate (y=2), the width should be 4 m, but the formula gives 4 m. At the bottom (y=3), the width should be 0 m, but the formula gives 0 m. Wait, let's re-evaluate. The plate is a triangle with top edge 4m and vertex 1m below. So height is 1m. Top is at y=2, bottom at y=3. Width at y=2 is 4. Width at y=3 is 0. The width varies linearly. w(y) = m*y + b. w(2)=4, w(3)=0. Slope m = (0-4)/(3-2) = -4. w(y) = -4(y-3) = 12 - 4y. The formula is actually correct. Let me re-read carefully. "A vertical plate shaped like a triangle with a 4-m horizontal top edge and its vertex 1 m below that edge". Top edge length = 4. Height = 1. Vertex is at the bottom. So it's an inverted triangle? No, "vertex 1 m below that edge" usually implies the vertex is the point opposite the base. If the top edge is the base, the vertex is at the bottom. So width goes from 4 to 0. My check: w(2) = 12 - 8 = 4. w(3) = 12 - 12 = 0. The formula is correct. Why did I think it was wrong? Let me check the integral. Integral from 2 to 3 of 9800*y*(12-4y) dy. 9800 * Integral (12y - 4y^2) dy. Antiderivative: 6y^2 - (4/3)y^3. Evaluated from 2 to 3. At 3: 6(9) - (4/3)(27) = 54 - 36 = 18. At 2: 6(4) - (4/3)(8) = 24 - 32/3 = 72/3 - 32/3 = 40/3. Result: 18 - 40/3 = 54/3 - 40/3 = 14/3. Force = 9800 * (14/3) = 137200/3. The calculation is correct. The setup is correct. Is there a subtle error? "vertex 1 m below that edge". If the triangle is oriented with the vertex at the top, the width would go from 0 to 4. But it says "4-m horizontal top edge", so the wide part is at the top. So it is an inverted triangle (base up). The width function is correct. The limits are correct (top 2m below surface, height 1m, so bottom is 3m below surface). The pressure is correct. The answer is correct. Why would this fail? Maybe the sentence is misleading? "A horizontal strip at depth y has width w(y) = 12 - 4*y". This is a correct statement. Is there any ambiguity? No. It seems sound. Wait, looking at the prompt again. "Review it." If it's correct, I should say pass. Let me double check the width derivation. Similar triangles. Let x be distance from the vertex. If vertex is at y=3, then distance from vertex is 3-y. Width is proportional to distance from vertex. w = k * (3-y). At y=2, w=4. 4 = k * (3-2) => k=4. So w(y) = 4(3-y) = 12 - 4y. Correct. Limits: y goes from 2 to 3. Correct. Pressure: 9800y. Correct. Integral setup: Correct. Result: Correct. Verdict: pass.gpt-oss:20b: pass 2026-10-06
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.