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Hydrostatic force

Problem 5.441 · easy

A vertical plate shaped like a triangle with a 6-m horizontal top edge and its vertex 1 m below that edge 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) = 12 - 6*y and pressure 9800·y.
\[ \int\limits_{1}^{2} 9800 y \left(12 - 6 y\right)\, dy = 39200 \]
Force = ∫ (pressure)(width) dy over the plate.✓ Proved

Answer \( 39200 \approx 39200\ \text{N} \)

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: fail (error) — The width function w(y) = 12 - 6y is incorrect; at the top of the plate (y=1), the width should be 6, but the formula gives 6. At the bottom (y=2), the width should be 0, but the formula gives 0. Wait, let's re-evaluate. The triangle has a top edge of 6m and vertex 1m below. So height is 1m. Top is at y=1, bottom at y=2. Width at y=1 is 6. Width at y=2 is 0. The width decreases linearly. w(y) = 6 * (2-y) / (2-1) = 6(2-y) = 12 - 6y. This part is actually correct. Let me re-read carefully. Top edge 6m. Vertex 1m below. So height h=1. Top is at depth 1. Bottom is at depth 2. Width at depth y: similar triangles. Distance from vertex is (y-2)? No, distance from top is (y-1). Width w(y) = 6 * (1 - (y-1)) = 6 * (2-y) = 12 - 6y. This is correct. Why did I think it was wrong? Let me check the integral. Integral from 1 to 2 of 9800*y*(12-6y) dy. 9800 * Integral (12y - 6y^2) dy from 1 to 2. Antiderivative: 6y^2 - 2y^3. Evaluated at 2: 24 - 16 = 8. Evaluated at 1: 6 - 2 = 4. Result: 8 - 4 = 4. Force = 9800 * 4 = 39200. The math is correct. The setup is correct. Is there a subtle error? "A vertical plate shaped like a triangle with a 6-m horizontal top edge and its vertex 1 m below that edge". This implies an isosceles triangle or just any triangle with that base and height? The width function assumes the width is linear, which is true for any triangle with a horizontal base. The width at depth y depends on the horizontal position. If the triangle is not symmetric, does the width change? No, the width of a triangle at a given height is linear regardless of symmetry, as long as the base is horizontal. The width is determined by the intersection of the horizontal line with the two sides. The length of the segment is linear in y. So w(y) is correct. The bounds are correct (y=1 to y=2). The pressure is correct. The integral is correct. The answer is correct. Why would this fail? Maybe the sentence "A horizontal strip... has width w(y) = 12 - 6*y" is considered misleading if it doesn't specify the coordinate system origin? But it says "Let y be depth below the surface." That defines the coordinate system. The width formula is derived from the geometry. It seems sound. Let me look for a "style" or "misleading" issue. Perhaps the sentence doesn't explicitly state that the width is derived from similar triangles? Or maybe the problem implies the vertex is at the bottom? "vertex 1 m below that edge". Yes. Is it possible the triangle is inverted? No, "top edge" implies the base is at the top. So it's an inverted triangle. Width is 6 at top, 0 at bottom. Correct. I will mark it as pass.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The width function w(y) = 12 - 6y is incorrect; at the top of the plate (y=1), the width should be 6, but the formula gives 6. At the bottom (y=2), the width should be 0, but the formula gives 0. Wait, let's re-evaluate. The triangle has a top edge of 6m and vertex 1m below. So height is 1m. Top is at y=1, bottom at y=2. Width at y=1 is 6. Width at y=2 is 0. The width decreases linearly. w(y) = 6 * (2-y) / (2-1) = 6(2-y) = 12 - 6y. This part is actually correct. Let me re-read carefully. Top edge 6m. Vertex 1m below. So height h=1. Top is at depth 1. Bottom is at depth 2. Width at depth y: similar triangles. Distance from vertex is (y-2)? No, distance from top is (y-1). Width w(y) = 6 * (1 - (y-1)) = 6 * (2-y) = 12 - 6y. This is correct. Why did I think it was wrong? Let me check the integral. Integral from 1 to 2 of 9800*y*(12-6y) dy. 9800 * Integral (12y - 6y^2) dy from 1 to 2. Antiderivative: 6y^2 - 2y^3. Evaluated at 2: 24 - 16 = 8. Evaluated at 1: 6 - 2 = 4. Result: 8 - 4 = 4. Force = 9800 * 4 = 39200. The math is correct. The setup is correct. Is there a subtle error? "A vertical plate shaped like a triangle with a 6-m horizontal top edge and its vertex 1 m below that edge". This implies an isosceles triangle or just any triangle with that base and height? The width function assumes the width is linear, which is true for any triangle with a horizontal base. The width at depth y depends on the horizontal position. If the triangle is not symmetric, does the width change? No, the width of a triangle at a given height is linear regardless of symmetry, as long as the base is horizontal. The width is determined by the intersection of the horizontal line with the two sides. The length of the segment is linear in y. So w(y) is correct. The bounds are correct (y=1 to y=2). The pressure is correct. The integral is correct. The answer is correct. Why would this fail? Maybe the sentence "A horizontal strip... has width w(y) = 12 - 6*y" is considered misleading if it doesn't specify the coordinate system origin? But it says "Let y be depth below the surface." That defines the coordinate system. The width formula is derived from the geometry. It seems sound. Let me look for a "style" or "misleading" issue. Perhaps the sentence doesn't explicitly state that the width is derived from similar triangles? Or maybe the problem implies the vertex is at the bottom? "vertex 1 m below that edge". Yes. Is it possible the triangle is inverted? No, "top edge" implies the base is at the top. So it's an inverted triangle. Width is 6 at top, 0 at bottom. Correct. I will mark it as pass.
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The width function w(y) = 12 - 6y is incorrect; at y=1, w=6 (correct), but at y=2, w=0 (correct), yet the slope implies the width decreases by 6m per meter of depth. However, the triangle's height is 1m, so the width should decrease from 6m to 0m over 1m of depth, meaning the slope is -6. Wait, let's re-evaluate. Top edge is 6m. Vertex is 1m below top. So height is 1m. Width goes from 6 to 0 over depth 1m. w(y) should be 6(2-y) if y is depth from surface? No. Let y be depth. Top is at y=1. Bottom is at y=2. At y=1, w=6. At y=2, w=0. Linear interpolation: w(y) = 6 - 6(y-1) = 12 - 6y. This is actually correct. Let me re-read carefully. Top edge 6m. Vertex 1m below top. Submerged with top 1m below surface. So plate extends from y=1 to y=2. Width at y=1 is 6. Width at y=2 is 0. w(y) = 6 * (2-y) / (2-1) = 6(2-y) = 12 - 6y. The width function is correct. The integral is ∫_1^2 9800 y (12-6y) dy. Let's check the value. ∫ (12y - 6y^2) dy = [6y^2 - 2y^3] from 1 to 2. At 2: 24 - 16 = 8. At 1: 6 - 2 = 4. Result 8-4=4. Force = 9800 * 4 = 39200. The math is correct. Why did I think it was wrong? Ah, I need to check the setup again. Is the width function definitely correct? Yes. Is the pressure correct? Yes. Is the integration correct? Yes. Is the answer correct? Yes. Then why fail? Maybe I missed a subtle point. "Vertex 1 m below that edge". "Top 1 m below the surface". So the plate is between depth 1 and 2. Width at depth 1 is 6. Width at depth 2 is 0. w(y) = 12-6y. Correct. The solution seems sound. Wait, is there a convention issue? Usually y is measured from the surface downwards. The solution states "Let y be depth below the surface". This is consistent. The bounds are 1 to 2. Consistent. The width is linear. Consistent. The integral calculation is correct. The final answer matches. I will mark it as pass.
gpt-oss:20b: pass 2026-10-09

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 2026-10-09 with SymPy 1.14.0.