Related rates
Problem 3.216 · medium
Car A drives east from an intersection at 38 mph and car B drives north from it at 54 mph. How fast is the distance between them changing when A is 5 mi and B is 12 mi from the intersection?
- With A at distance a and B at distance b, the distance between them satisfies D² = a² + b².Reviewed
- Differentiate: 2D dD/dt = 2a da/dt + 2b db/dt.Reviewed
- \[ \left. \sqrt{x^{2} + y^{2}} \right|_{\substack{ x=5\\ y=12 }} = 13 \]D at that moment.✓ Proved
- \[ \left. \frac{38 x + 54 y}{\sqrt{x^{2} + y^{2}}} \right|_{\substack{ x=5\\ y=12 }} = \frac{838}{13} \]dD/dt = (a·da/dt + b·db/dt)/D with a = 5, b = 12.✓ Proved
Answer \( \frac{838}{13} \ \text{mph} \approx 64.46\ \text{mph} \)
✓ 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 using the Pythagorean theorem, differentiates implicitly, and substitutes the given values accurately.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly sets up the related rates problem using the Pythagorean theorem, differentiates implicitly, and substitutes the given values accurately.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly sets up the related rates problem using the Pythagorean theorem, differentiates implicitly, and substitutes the correct values for position and velocity to arrive at the correct answer.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.