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Hydrostatic force

Problem 5.302 · easy

A vertical plate shaped like a triangle with a 3-ft horizontal top edge and its vertex 1 ft below that edge is submerged in water with its top 3 ft below the surface. Find the force of the water on one side. (Water weighs \( \displaystyle \frac{312}{5}\ \text{lb}/\text{ft}^3 \).)

Let y be depth below the surface. A horizontal strip at depth y has width w(y) = 12 - 3*y and pressure 312/5·y.
Reviewed
\[ \int\limits_{3}^{4} \frac{312 y \left(12 - 3 y\right)}{5}\, dy = 312 \]
Force = ∫ (pressure)(width) dy over the plate.✓ Proved

Answer \( 312 \approx 312\ \text{lb} \)

Lines: 1 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly models the geometry and physics. The width function w(y) = 12 - 3y is derived correctly from the triangle's dimensions (width 3 at depth 3, width 0 at depth 4). The pressure formula and integration bounds are correct, leading to the right answer.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly models the geometry and physics. The width function w(y) = 12 - 3y is derived correctly from the triangle's dimensions (width 3 at depth 3, width 0 at depth 4). The pressure formula and integration bounds are correct, leading to the right answer.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: fail (error) 2026-10-06 — The width function w(y) = 12 - 3y is incorrect; at the top of the plate (y=3), the width should be 3 ft, but the formula gives 3 ft, while at the bottom (y=4), the width should be 0, but the formula gives 0. Wait, let's re-evaluate. At y=3, w=12-9=3. At y=4, w=12-12=0. The width function is actually correct for the geometry described (top width 3, bottom width 0, height 1). However, the pressure term is missing the density factor in the sentence description? No, the sentence says pressure is 312/5 * y. The integral uses 312/5 * y * w(y). The integral limits are 3 to 4. Let's check the integral value. Integral of (312/5)(12y - 3y^2) from 3 to 4. Antiderivative: (312/5)(6y^2 - y^3). Evaluated at 4: 6(16)-64 = 96-64=32. Evaluated at 3: 6(9)-27 = 54-27=27. Difference: 32-27=5. Result: (312/5)*5 = 312. The calculation is correct. Why did I think it was wrong? Let me re-read the width derivation. Triangle with top edge 3ft, vertex 1ft below. Let x be distance from top. Width w(x) = 3 - 3x (since it goes from 3 to 0 over length 1). Depth y = 3 + x. So x = y - 3. w(y) = 3 - 3(y-3) = 3 - 3y + 9 = 12 - 3y. The width function is correct. The limits are correct (top at y=3, bottom at y=4). The density is correct. The result is correct. Is there a subtle error? "A horizontal strip... has width w(y)... and pressure...". The force on a strip is pressure * area. Area = width * dy. The integral sums pressure * width * dy. This is correct. The solution seems sound. Wait, looking closer at the prompt's request for adversarial review. Is there anything misleading? The sentence says "pressure 312/5·y". This is gauge pressure, which is standard for hydrostatic force problems unless atmospheric pressure is specified to act on both sides (which cancels out). The problem asks for "force of the water", which implies gauge pressure. The setup is correct. The calculation is correct. The answer is correct. I will mark it as pass.

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.