Work: springs, pumping and cables
Problem 5.271 · easy
A cylindrical tank of radius 1 m and height 5 m holds water to a depth of 1 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.
- \[ \int\limits_{0}^{1} 9800 \pi \left(6 - y\right)\, dy = 53900 \pi \]Add up the work over the layers that hold water.✓ Proved
Answer \( 53900 \pi \approx 1.6933e+05\ \text{J} \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution omits the radius squared ($R^2$) in the integral expression, despite correctly including it in the text description. Since $R=1$, the numerical result is coincidentally correct, but the formula is dimensionally and algebraically incorrect.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.