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Work: springs, pumping and cables

Problem 5.270 · easy

A 40-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.
\[ \int\limits_{0}^{40} x\, dx = 800 \]
Add it up.✓ Proved

Answer \( 800\ \text{ft-lb} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution omits the weight density (1 lb/ft) from the integrand, calculating the integral of x instead of 1*x. This yields 800 instead of the correct 800, but for the wrong reason (missing the physical quantity in the formula).

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution omits the weight density (1 lb/ft) from the integrand, calculating the integral of x instead of 1*x. This yields 800 instead of the correct 800, but for the wrong reason (missing the physical quantity in the formula).
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution omits the linear density (1 lb/ft) in the integral setup, effectively assuming a density of 1 without stating it, and fails to define the integrand explicitly before integrating.
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.