Work: springs, pumping and cables
Problem 5.546 · easy
A 10-ft cable weighing 1 lb/ft hangs from the top of a building. How much work does it take to pull it all to the top?
- Measure x down from the top. The piece of cable at depth x weighs ρ·dx and rises x feet.Reviewed
- \[ \int\limits_{0}^{10} x\, dx = 50 \]Add it up.✓ Proved
Answer \( 50\ \text{ft-lb} \)
Lines: 1 proved, 1 reviewed. 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 | 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: passqwen3.6:27b-mlx: pass — The setup correctly identifies the weight of the differential element and the distance it must be lifted. The integration bounds and result are correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The setup correctly identifies the weight of the differential element and the distance it must be lifted. The integration bounds and result are correct.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution omits the linear density (1 lb/ft) from the integrand, calculating the integral of x instead of 1*x. This results in a numerical coincidence rather than a correct physical derivation.
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 2026-10-13 with SymPy 1.14.0.