Related rates
Problem 3.217 · medium
A 10 ft ladder leans against a vertical wall. The bottom slides away from the wall at 2 ft/s. How fast is the top sliding down the wall when the bottom is 6 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 = 100.Reviewed
- Differentiate with respect to t: 2x dx/dt + 2y dy/dt = 0, so dy/dt = -(x/y) dx/dt.Reviewed
- \[ \left. \sqrt{100 - x^{2}} \right|_{\substack{ x=6 }} = 8 \]When x = 6, y = √(100 − 36).✓ Proved
- \[ \left. - \frac{v x}{y} \right|_{\substack{ x=6\\ y=8\\ v=2 }} = - \frac{3}{2} \]dy/dt = −(x/y)(dx/dt) with x = 6, y = 8, dx/dt = 2.✓ Proved
Answer \( - \frac{3}{2} \ \text{ft/s} \approx -1.5\ \text{ft/s} \)
✓ 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 models the related rates problem, applies implicit differentiation properly, and substitutes the correct values to arrive at the right answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly models the related rates problem, applies implicit differentiation properly, and substitutes the correct values to arrive at the right answer.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly sets up the related rates problem, applies implicit differentiation, and substitutes the correct values to find the rate.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.