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Work: springs, pumping and cables

Problem 5.335 · easy

A cylindrical tank of radius 1 m and height 5 m holds water to a depth of 5 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 5 − y meters.
\[ \int\limits_{0}^{5} 9800 \pi \left(5 - y\right)\, dy = 122500 \pi \]
Add up the work over the layers that hold water.✓ Proved

Answer \( 122500 \pi \approx 3.8485e+05\ \text{J} \)

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 = 1^2) from the integrand in the equation, although it was correctly included in the text description. While the numerical result is coincidentally correct because R=1, the equation as written is algebraically incorrect for the general case and fails to model the volume of the slice properly.

Every verdict on record (4)

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) from the integrand in the equation, although it was correctly included in the text description. While the numerical result is coincidentally correct because R=1, the equation as written is algebraically incorrect for the general case and fails to model the volume of the slice properly.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution omits the radius squared term (R^2 = 1^2) in the integral setup, although it is implicitly used in the final calculation. The sentence states the weight is 9800*pi*R^2*dy, but the integral line only includes 9800*pi*(5-y), missing the R^2 factor explicitly, which is a significant omission in the modeling step.

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.