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Hydrostatic force

Problem 5.364 · easy

A vertical plate shaped like a triangle with a 4-m horizontal top edge and its vertex 3 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) = 8 - 4*y/3 and pressure 9800·y.
\[ \int\limits_{3}^{6} 9800 y \left(8 - \frac{4 y}{3}\right)\, dy = 235200 \]
Force = ∫ (pressure)(width) dy over the plate.✓ Proved

Answer \( 235200 \approx 235200\ \text{N} \)

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: pass
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-object
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The width function w(y) is incorrect. At y=3 (top of plate), the width should be 4, but the formula gives 8 - 4 = 4 (correct). At y=6 (bottom vertex), the width should be 0, but the formula gives 8 - 8 = 0 (correct). Wait, let's re-evaluate. The triangle has a top edge of 4m and height 3m. The vertex is 3m below the top edge. The plate is submerged with its top 3m below the surface. So the plate extends from y=3 to y=6. The width varies linearly from 4 at y=3 to 0 at y=6. The slope is (0-4)/(6-3) = -4/3. The equation of the line for width is w(y) - 0 = -4/3 * (y - 6) => w(y) = -4/3(y-6) = -4/3y + 8 = 8 - 4y/3. This matches the solution's width function. Let's check the integral. Integral from 3 to 6 of 9800 * y * (8 - 4y/3) dy. Let's calculate: 9800 * Integral(8y - 4/3 y^2) dy from 3 to 6. Antiderivative: 4y^2 - 4/9 y^3. Evaluated at 6: 4(36) - 4/9(216) = 144 - 96 = 48. Evaluated at 3: 4(9) - 4/9(27) = 36 - 12 = 24. Difference: 48 - 24 = 24. Total force: 9800 * 24 = 235200. The calculation is correct. Why did I think it was wrong? Let me re-read the problem carefully. "A vertical plate shaped like a triangle with a 4-m horizontal top edge and its vertex 3 m below that edge is submerged in water with its top 3 m below the surface." Top edge width = 4. Height of triangle = 3. Top of plate is at depth 3. Bottom of plate (vertex) is at depth 3+3=6. Width at depth y: w(y). At y=3, w=4. At y=6, w=0. Linear interpolation: w(y) = 4 * (6-y)/3 = 4/3 * (6-y) = 8 - 4y/3. This is correct. The integral setup is correct. The result is correct. Is there a subtle error? "Find the force of the water on one side." The solution calculates the force. The answer matches. Is the width function definitely correct? Yes. Is the pressure correct? Yes, 9800*y. Is the integration variable correct? Yes, dy. Are the bounds correct? Yes, 3 to 6. It seems correct. Let me double check the "adversarial" part. Is there a convention issue? Sometimes y is measured from the top of the plate. Here y is depth below surface. This is standard. The width function derivation is sound. The integral evaluation is sound. The final answer is sound. I will mark it as pass.
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.