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Work: springs, pumping and cables

Problem 5.338 · easy

A cylindrical tank of radius 1 m and height 5 m holds water to a depth of 2 m. How much work does it take to pump all the water out through a spout 1 m above the top? (Water weighs \( \displaystyle 9800\ \text{N/m}^3 \).)

Slice the water into horizontal layers of thickness dy at height y. Each weighs 9800·π·R²·dy and must rise 6 − y meters.
Reviewed
\[ \int\limits_{0}^{2} 9800 \pi \left(6 - y\right)\, dy = 98000 \pi \]
Add up the work over the layers that hold water.✓ Proved

Answer \( 98000 \pi \approx 3.0788e+05\ \text{J} \)

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 setup correctly identifies the radius R=1, the lift distance (6-y), and the integration limits (0 to 2) for the water depth. The calculation is correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The setup correctly identifies the radius R=1, the lift distance (6-y), and the integration limits (0 to 2) for the water depth. The calculation is correct.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution omits the radius squared (R^2 = 1^2) in the integral expression, although it was included in the text description. While the numerical result is correct because R=1, the equation as written is dimensionally and algebraically incorrect for the general case described.

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/work, checked 2026-10-07 with SymPy 1.14.0.