Related rates
Problem 3.336 · medium
A 6 ft tall person walks away from a 18 ft lamppost at 3 ft/s. How fast is the length of their shadow increasing?
- Let x be the person's distance from the post and s the shadow's length. Similar triangles: 18/(x + s) = 6/s, so s = 6x/12.Reviewed
- \[ \frac{d}{d x} \frac{x}{2} = \frac{1}{2} \]ds/dt = (ds/dx)(dx/dt).✓ Proved
- \[ \frac{3}{2} \]Multiply by dx/dt = 3.✓ Proved
Answer \( \frac{3}{2} \ \text{ft/s} \approx 1.5\ \text{ft/s} \)
✓ Nihil obstat Lines: 2 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 |
| 3 | ✓ 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 similar triangles relationship, derives the linear relationship between shadow length and distance, and correctly applies the chain rule to find the rate of change.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly sets up the similar triangles relationship, derives the linear relationship between shadow length and distance, and correctly applies the chain rule to find the rate of change.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly sets up the similar triangles relationship, derives the linear relationship between shadow length and distance, and correctly applies the chain rule 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.