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Work: springs, pumping and cables

Problem 5.334 · easy

A cylindrical tank of radius 1 m and height 6 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 7 − y meters.
Reviewed
\[ \int\limits_{0}^{2} 9800 \pi \left(7 - y\right)\, dy = 117600 \pi \]
Add up the work over the layers that hold water.✓ Proved

Answer \( 117600 \pi \approx 3.6945e+05\ \text{J} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

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 sets up the work integral by defining the slice weight and lift distance relative to the chosen coordinate system. The bounds and constants are applied correctly.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly sets up the work integral by defining the slice weight and lift distance relative to the chosen coordinate system. The bounds and constants are applied correctly.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The setup correctly identifies the volume of a slice as πR²dy (with R=1) and the distance to lift as (7-y), integrating from y=0 to y=2. The calculation is correct.

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.