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Work: springs, pumping and cables

Problem 5.407 · easy

A 40-ft cable weighing 3 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.
Reviewed
\[ \int\limits_{0}^{40} \left(3 x + 10\right)\, dx = 2800 \]
Add it up.✓ Proved

Answer \( 2800\ \text{ft-lb} \)

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: pass
qwen3.6:27b-mlx: pass — The setup correctly models the work done by integrating the force required to lift each segment of the cable and the load over their respective distances. The integral calculation is correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The setup correctly models the work done by integrating the force required to lift each segment of the cable and the load over their respective distances. The integral calculation is correct.
gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The integrand is incorrect. The work to lift the load is constant force times distance (10 * 40 = 400 ft-lb), not integrated as 10 dx over the length. The cable work is correct (integral of 3x dx), but the load term is modeled 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 2026-10-09 with SymPy 1.14.0.