Related rates
Problem 3.338 · medium
A 25 ft ladder leans against a vertical wall. The bottom slides away from the wall at 1 ft/s. How fast is the top sliding down the wall when the bottom is 7 ft from the wall?
- Let x be the distance from the wall to the bottom and y the height of the top. The ladder is the hypotenuse: x² + y² = 625.Reviewed
- Differentiate with respect to t: 2x dx/dt + 2y dy/dt = 0, so dy/dt = -(x/y) dx/dt.Reviewed
- \[ \left. \sqrt{625 - x^{2}} \right|_{\substack{ x=7 }} = 24 \]When x = 7, y = √(625 − 49).✓ Proved
- \[ \left. - \frac{v x}{y} \right|_{\substack{ x=7\\ y=24\\ v=1 }} = - \frac{7}{24} \]dy/dt = −(x/y)(dx/dt) with x = 7, y = 24, dx/dt = 1.✓ Proved
Answer \( - \frac{7}{24} \ \text{ft/s} \approx -0.2917\ \text{ft/s} \)
✓ Nihil obstat Lines: 2 proved, 2 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ 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 | the situation was stepped forward and back by a microsecond and the quantity differenced numerically |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly sets up the related rates problem, applies the Pythagorean theorem and implicit differentiation, and substitutes the correct values to find the rate.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly sets up the related rates problem, applies the Pythagorean theorem and implicit differentiation, and substitutes the correct values to find the rate.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly sets up the related rates problem, applies implicit differentiation, and substitutes the correct values to find the rate of change.gpt-oss:20b: pass 2026-10-03
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/related_rates, checked 2026-10-03 with SymPy 1.14.0.