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Hydrostatic force

Problem 5.363 · easy

A vertical plate shaped like a triangle with its vertex at the top and a 3-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) = 3*y - 3 and pressure 9800·y.
\[ \int\limits_{1}^{2} 9800 y \left(3 y - 3\right)\, dy = 24500 \]
Force = ∫ (pressure)(width) dy over the plate.✓ Proved

Answer \( 24500 \approx 24500\ \text{N} \)

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) = 3y - 3 is incorrect; at the top of the plate (y=1), the width should be 0, but this formula gives 0, which is correct, but at the bottom (y=2), the width should be 3, and 3(2)-3 = 3, which is also correct. Wait, let's re-evaluate. The vertex is at y=1 (1m below surface). The base is at y=2 (1m below vertex, so 2m below surface). The width at depth y is proportional to the distance from the vertex. Distance from vertex is (y-1). At y=2, distance is 1, width is 3. So w(y) = 3(y-1) = 3y - 3. This part is actually correct. Let me re-read carefully. Vertex at top. Base 1m below vertex. Top 1m below surface. So vertex is at depth 1m. Base is at depth 2m. Width at depth y: similar triangles. Width/3 = (y-1)/1 => w(y) = 3(y-1). This matches 3y-3. The integral is from y=1 to y=2. Pressure is 9800y. Integral of 9800y(3y-3) from 1 to 2. Let's check the value. 9800 * integral(3y^2 - 3y) dy from 1 to 2. Antiderivative: y^3 - 1.5y^2. Evaluated at 2: 8 - 6 = 2. Evaluated at 1: 1 - 1.5 = -0.5. Difference: 2 - (-0.5) = 2.5. 9800 * 2.5 = 24500. The calculation is correct. Why did I think it was wrong? Ah, I need to check the setup description again. "A horizontal strip at depth y has width w(y) = 3*y - 3". This is correct. "Pressure 9800*y". Correct. Bounds 1 to 2. Correct. Result 24500. Correct. Is there a subtle error? "Vertex at the top". "Base 1 m below the vertex". "Top 1 m below the surface". So the triangle extends from depth 1 to depth 2. The width at depth y is indeed 3(y-1). The sentence says w(y) = 3y - 3. This is correct. The integral is correct. The answer is correct. Wait, is the unit correct? N/m^3 * m * m * m = N. Yes. Is there any ambiguity? "Force of the water on one side". Yes. It seems correct. Let me look for a trick. Maybe the orientation? "Vertical plate... vertex at the top". Standard setup. Maybe the width formula derivation is missing? The sentence just states it. Is it misleading? No, it's a standard result. Is it an error? No. Pass?

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The width function w(y) = 3y - 3 is incorrect; at the top of the plate (y=1), the width should be 0, but this formula gives 0, which is correct, but at the bottom (y=2), the width should be 3, and 3(2)-3 = 3, which is also correct. Wait, let's re-evaluate. The vertex is at y=1 (1m below surface). The base is at y=2 (1m below vertex, so 2m below surface). The width at depth y is proportional to the distance from the vertex. Distance from vertex is (y-1). At y=2, distance is 1, width is 3. So w(y) = 3(y-1) = 3y - 3. This part is actually correct. Let me re-read carefully. Vertex at top. Base 1m below vertex. Top 1m below surface. So vertex is at depth 1m. Base is at depth 2m. Width at depth y: similar triangles. Width/3 = (y-1)/1 => w(y) = 3(y-1). This matches 3y-3. The integral is from y=1 to y=2. Pressure is 9800y. Integral of 9800y(3y-3) from 1 to 2. Let's check the value. 9800 * integral(3y^2 - 3y) dy from 1 to 2. Antiderivative: y^3 - 1.5y^2. Evaluated at 2: 8 - 6 = 2. Evaluated at 1: 1 - 1.5 = -0.5. Difference: 2 - (-0.5) = 2.5. 9800 * 2.5 = 24500. The calculation is correct. Why did I think it was wrong? Ah, I need to check the setup description again. "A horizontal strip at depth y has width w(y) = 3*y - 3". This is correct. "Pressure 9800*y". Correct. Bounds 1 to 2. Correct. Result 24500. Correct. Is there a subtle error? "Vertex at the top". "Base 1 m below the vertex". "Top 1 m below the surface". So the triangle extends from depth 1 to depth 2. The width at depth y is indeed 3(y-1). The sentence says w(y) = 3y - 3. This is correct. The integral is correct. The answer is correct. Wait, is the unit correct? N/m^3 * m * m * m = N. Yes. Is there any ambiguity? "Force of the water on one side". Yes. It seems correct. Let me look for a trick. Maybe the orientation? "Vertical plate... vertex at the top". Standard setup. Maybe the width formula derivation is missing? The sentence just states it. Is it misleading? No, it's a standard result. Is it an error? No. Pass?
qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-object
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.