Work: springs, pumping and cables
Problem 5.473 · easy
A 40-ft cable weighing 1 lb/ft hangs from the top of a building with a 10-lb load on its end. 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; the load rises the full length.
- \[ \int\limits_{0}^{40} \left(x + 10\right)\, dx = 1200 \]Add it up.✓ Proved
Answer \( 1200\ \text{ft-lb} \)
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: fail (error) — The integrand is missing the linear density factor (1 lb/ft). The work for the cable segment is 1*x*dx, not x*dx. The correct integral is ∫(x + 10)dx, which evaluates to 1200, but the setup in line 1 implies the weight is just dx, which is dimensionally and physically incorrect.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The integrand is missing the linear density factor (1 lb/ft). The work for the cable segment is 1*x*dx, not x*dx. The correct integral is ∫(x + 10)dx, which evaluates to 1200, but the setup in line 1 implies the weight is just dx, which is dimensionally and physically incorrect.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The integrand is incorrect; it sums the position x and the load weight 10, rather than multiplying the force (weight of cable segment + load) by the distance moved. The correct integrand for the cable part is x * dx, and for the load is 10 * 40, or combined as (x + 10) * dx if x represents the weight of the cable segment at that height, but the setup implies x is distance. Specifically, the work for the cable is integral of x * 1 dx from 0 to 40, and for the load is 10 * 40. The expression Integral(x + 10, ...) suggests adding distance and weight, which is dimensionally inconsistent and mathematically wrong.
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 2026-10-11 with SymPy 1.14.0.