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Work: springs, pumping and cables

Problem 5.477 · easy

A cylindrical tank of radius 1 m and height 4 m holds water to a depth of 1 m. How much work does it take to pump all the water out over 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 4 − y meters.
\[ \int\limits_{0}^{1} 9800 \pi \left(4 - y\right)\, dy = 34300 \pi \]
Add up the work over the layers that hold water.✓ Proved

Answer $ 34300 \pi \approx 1.0776e+05\ \text{J} $

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 solution omits the radius squared ($R^2$) in the integral, despite including it in the text description. With $R=1$, the numerical result is coincidentally correct, but the formula is dimensionally and mathematically incorrect.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution omits the radius squared ($R^2$) in the integral, despite including it in the text description. With $R=1$, the numerical result is coincidentally correct, but the formula is dimensionally and mathematically incorrect.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution omits the radius squared (R^2 = 1) in the integral expression, although the numerical result is correct because R=1. This is a significant algebraic error in the setup that would lead to incorrect results for any other radius.

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 2026-10-11 with SymPy 1.14.0.