Home›Calculus 2›Work: springs, pumping and cables›Problem 5.265

Work: springs, pumping and cables

Problem 5.265 · easy

A 30-ft cable weighing 3 lb/ft hangs from the top of a building with a 25-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
\[ \int\limits_{0}^{30} \left(3 x + 25\right)\, dx = 2100 \]
Add it up.✓ Proved

Answer \( 2100\ \text{ft-lb} \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

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 solution correctly models the work done by integrating the force required to lift each segment of the cable and the load over the distance they are lifted. The setup and calculation are sound.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly models the work done by integrating the force required to lift each segment of the cable and the load over the distance they are lifted. The setup and calculation are sound.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly sets up the work integral by summing the work done on each differential segment of the cable and the constant load. The algebraic evaluation is correct.
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.