Hydrostatic force
Problem 5.294 · easy
A vertical plate shaped like a triangle with a 4-m horizontal top edge and its vertex 4 m below that edge is submerged in water with its top 3 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) = 7 - y and pressure 9800·y.
- \[ \int\limits_{3}^{7} 9800 y \left(7 - y\right)\, dy = \frac{1019200}{3} \]Force = ∫ (pressure)(width) dy over the plate.✓ Proved
Answer \( \frac{1019200}{3} \approx 339733\ \text{N} \)
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) = 7 - y is incorrect; for a triangle with a 4m top edge at depth 3m and vertex at depth 7m, the width should be w(y) = 2(y - 3). The solution incorrectly assumes a width of 4 at the top (y=3) implies a slope that doesn't match the geometry, or simply guesses the linear function without deriving it from the similar triangles.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The width function w(y) = 7 - y is incorrect; for a triangle with a 4m top edge at depth 3m and vertex at depth 7m, the width should be w(y) = 2(y - 3). The solution incorrectly assumes a width of 4 at the top (y=3) implies a slope that doesn't match the geometry, or simply guesses the linear function without deriving it from the similar triangles.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The width function w(y) = 7 - y is incorrect; at the top of the plate (y=3), the width should be 4, but the formula gives 4. At the bottom (y=7), the width should be 0, and the formula gives 0. However, the linear relationship is wrong. The width decreases from 4 to 0 over a depth of 4m. The correct width is w(y) = 4 * (7-y)/4 = 7-y? No. Let's check the geometry. Top edge is at y=3, width 4. Vertex is at y=7, width 0. The width is linear. w(y) = m(y-3) + 4. w(7) = 0 => 4m + 4 = 0 => m = -1. So w(y) = -(y-3) + 4 = -y + 3 + 4 = 7 - y. Wait, the width function is actually correct. Let me re-read carefully. Top edge 4m horizontal. Vertex 4m below that edge. Top is 3m below surface. So plate extends from y=3 to y=7. Width at y=3 is 4. Width at y=7 is 0. Linear interpolation: w(y) = 4 * (7-y)/(7-3) = 4 * (7-y)/4 = 7-y. The width function is correct. The integral limits are 3 to 7. The pressure is 9800*y. The integral is correct. Why did I think it was wrong? Let me re-evaluate. Is there a subtle error? The problem asks for force on one side. The setup seems correct. Let me check the calculation. Integral of 9800(7y - y^2) from 3 to 7. Antiderivative: 9800(7y^2/2 - y^3/3). Evaluated at 7: 9800(49*7/2 - 343/3) = 9800(343/2 - 343/3) = 9800(343/6). Evaluated at 3: 9800(7*9/2 - 27/3) = 9800(63/2 - 9) = 9800(63/2 - 18/2) = 9800(45/2). Difference: 9800(343/6 - 135/6) = 9800(208/6) = 9800(104/3) = 1019200/3. The math is correct. The width function is correct. The limits are correct. The pressure is correct. The answer matches. So why would it fail? Maybe the prompt implies I should find an error? Or maybe I missed something. Let's look at the sentence again. "A horizontal strip at depth y has width w(y) = 7 - y". This is correct. "pressure 9800·y". This is correct. "Integral... over the plate". This is correct. Is it possible the vertex is above the top edge? "vertex 4 m below that edge". No. Is the top edge 4m wide? Yes. Is the top 3m below surface? Yes. Everything seems correct. Wait, is the triangle inverted? "vertical plate shaped like a triangle with a 4-m horizontal top edge and its vertex 4 m below that edge". This describes an inverted triangle (point down). My width calculation assumed this. If it were point up, the top edge would be the base, but the vertex would be above? No, "vertex 4m below that edge" confirms point down. So the setup is correct. The solution is correct. I will mark it as 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.