Hydrostatic force
Problem 5.232 · easy
A vertical plate shaped like a triangle with its vertex at the top and a 5-m horizontal base 1 m below the vertex is submerged in water with its top 1 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) = 5*y - 5 and pressure 9800·y.
- \[ \int\limits_{1}^{2} 9800 y \left(5 y - 5\right)\, dy = \frac{122500}{3} \]Force = ∫ (pressure)(width) dy over the plate.✓ Proved
Answer \( \frac{122500}{3} \approx 40833.3\ \text{N} \)
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: fail (error) — The width of the plate at depth y is not 5y‑5. The triangle has width 0 at the top (y=1) and width 5 at the base (y=2), so w(y)=5(y‑1). Using 5y‑5 gives a negative width for y<1 and an incorrect force. The integral limits and width function must be corrected.qwen3.6:27b-mlx: pass — The setup correctly models the geometry (width linearly increasing from 0 at y=1 to 5 at y=2) and the physics (pressure proportional to depth). The integration bounds and result are correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The setup correctly models the geometry (width linearly increasing from 0 at y=1 to 5 at y=2) and the physics (pressure proportional to depth). The integration bounds and result are correct.gpt-oss:20b: fail (error) 2026-10-04 — The width of the plate at depth y is not 5y‑5. The triangle has width 0 at the top (y=1) and width 5 at the base (y=2), so w(y)=5(y‑1). Using 5y‑5 gives a negative width for y<1 and an incorrect force. The integral limits and width function must be corrected.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The width function w(y) = 5y - 5 is incorrect. At the surface (y=1), the width should be 0, but the formula gives 0. At y=2, the width should be 5, and the formula gives 5. However, the problem states the vertex is at the top, 1m below the surface. So at y=1, width is 0. At y=2, width is 5. The linear relationship is w(y) = 5(y-1). The solution uses w(y) = 5y - 5, which is actually 5(y-1). Wait, 5y - 5 is 5(y-1). So the width function is correct. Let me re-read carefully. Vertex at top. Top is 1m below surface. Base is 5m wide, 1m below vertex. So base is at depth 2m. Width at depth y: w(y) = (5/1)*(y-1) = 5(y-1) = 5y - 5. This is correct. The integral bounds are y=1 to y=2. This is correct. The pressure is 9800*y. This is correct. The integral is ∫(1 to 2) 9800*y*(5y-5) dy. Let's check the calculation. 9800 * ∫(1 to 2) (5y^2 - 5y) dy = 9800 * [5/3 y^3 - 5/2 y^2] from 1 to 2. At 2: 5/3(8) - 5/2(4) = 40/3 - 10 = 10/3. At 1: 5/3 - 5/2 = 10/6 - 15/6 = -5/6. Difference: 10/3 - (-5/6) = 20/6 + 5/6 = 25/6. Force = 9800 * 25/6 = 4900 * 25 / 3 = 122500 / 3. The calculation is correct. Why did I think it was wrong? Let me re-read the prompt's solution text. "w(y) = 5*y - 5". This is correct. "pressure 9800·y". Correct. Bounds 1 to 2. Correct. Result 122500/3. Correct. Is there a subtle error? "A horizontal strip at depth y has width w(y) = 5*y - 5". This implies the width is 0 at y=1 and 5 at y=2. This matches the geometry. The setup seems perfectly correct. Wait, is the vertex at the top? Yes. Is the base 1m below the vertex? Yes. Is the top 1m below the surface? Yes. So the plate extends from y=1 to y=2. The width at y=1 is 0. The width at y=2 is 5. The width function is linear. w(y) = 5(y-1). This is 5y-5. Everything seems correct. Let me look for a 'misleading' aspect. Maybe the variable y? Usually y is measured from the surface downwards. The solution says "Let y be depth below the surface." This is standard. The pressure is rho*g*y. Correct. The area element is w(y)dy. Correct. The integral is correct. The answer is correct. Is there any ambiguity? "Find the force of the water on one side." Standard hydrostatic force. I see no error. 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.