Hydrostatic force
Problem 5.229 · 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 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) = 15/2 - 5*y/4 and pressure 9800·y.
- \[ \int\limits_{2}^{6} 9800 y \left(\frac{15}{2} - \frac{5 y}{4}\right)\, dy = \frac{980000}{3} \]Force = ∫ (pressure)(width) dy over the plate.✓ Proved
Answer \( \frac{980000}{3} \approx 326667\ \text{N} \)
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: fail (error) — The width function w(y) is incorrect. At the top of the plate (y=2), the width should be 5, but the formula gives 15/2 - 5(2)/4 = 5.5. The correct width is w(y) = 5(y-6)/4, which is 5 at y=2 and 0 at y=6.
Every verdict on record (4)
qwen3.6:27b-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 5, but the formula gives 15/2 - 5(2)/4 = 5.5. The correct width is w(y) = 5(y-6)/4, which is 5 at y=2 and 0 at y=6.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The width function w(y) is incorrect. At y=2 (top), w should be 5, but the formula gives 15/2 - 10/4 = 5. At y=6 (bottom), w should be 0, but the formula gives 15/2 - 30/4 = 0. Wait, let's re-evaluate. Top is at y=2, width 5. Bottom is at y=6, width 0. The width varies linearly. w(y) = m*y + b. w(2)=5, w(6)=0. Slope m = (0-5)/(6-2) = -5/4. w(y) = -5/4(y-6) = -5/4 y + 30/4 = -5/4 y + 15/2. The formula in the solution is 15/2 - 5*y/4, which matches. So the width function is correct. Let's check the integral. Integral from 2 to 6 of 9800*y*(15/2 - 5*y/4) dy. Let's compute: 9800 * Integral(15/2 y - 5/4 y^2) dy from 2 to 6. Antiderivative: 15/4 y^2 - 5/12 y^3. Evaluate at 6: 15/4(36) - 5/12(216) = 15*9 - 5*18 = 135 - 90 = 45. Evaluate at 2: 15/4(4) - 5/12(8) = 15 - 10/3 = 35/3. Result: 45 - 35/3 = 135/3 - 35/3 = 100/3. Force = 9800 * 100/3 = 980000/3. The calculation is correct. Why did I think it was wrong? Let me re-read the prompt. "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 2 m below the surface." Top edge is 5m wide. Vertex is 4m below top edge. So height of triangle is 4m. Top is 2m below surface. Bottom is 2+4=6m below surface. Width at depth y: At y=2, w=5. At y=6, w=0. Linear interpolation: w(y) = 5 * (6-y)/4 = 5/4 * (6-y) = 30/4 - 5/4 y = 15/2 - 5/4 y. This matches the solution's width function. The integral limits are 2 to 6. This matches. The algebra is checked by the CAS. The setup is correct. The answer is correct. So why would it fail? Maybe I missed something subtle. "Find the force of the water on one side." Standard hydrostatic force problem. Pressure = rho * g * depth. Depth is y. Pressure = 9800 * y. Force = Integral Pressure * dA. dA = w(y) dy. Everything seems correct. Is there a convention issue? Sometimes y is measured from the top of the plate. If y was measured from the top of the plate, depth would be y+2. But the solution explicitly says "Let y be depth below the surface." So the variable definition is clear. The width function is derived correctly based on that variable. The limits are correct. The integral is correct. The result is correct. I will mark it as pass.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.