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Work: springs, pumping and cables

Problem 5.333 · easy

A cylindrical tank of radius 2 m and height 2 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 3 − y meters.
Reviewed
\[ \int\limits_{0}^{2} 39200 \pi \left(3 - y\right)\, dy = 156800 \pi \]
Add up the work over the layers that hold water.✓ Proved

Answer \( 156800 \pi \approx 4.926e+05\ \text{J} \)

✓ 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 setup correctly identifies the weight of a slice as density times area times thickness, and the distance to lift as (3 - y) given the spout is 1m above the 2m high tank. The integration bounds and result are 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 weight of a slice as density times area times thickness, and the distance to lift as (3 - y) given the spout is 1m above the 2m high tank. The integration bounds and result are correct.
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 weight of each slice and the distance it must be lifted (from height y to 3 m). The integration bounds and calculation are 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.