Related rates
Problem 3.255 · medium
Car A drives east from an intersection at 22 mph and car B drives north from it at 41 mph. How fast is the distance between them changing when A is 9 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
- \[ \left. \sqrt{x^{2} + y^{2}} \right|_{\substack{ x=9\\ y=12 }} = 15 \]D at that moment.✓ Proved
- \[ \left. \frac{22 x + 41 y}{\sqrt{x^{2} + y^{2}}} \right|_{\substack{ x=9\\ y=12 }} = 46 \]dD/dt = (a·da/dt + b·db/dt)/D with a = 9, b = 12.✓ Proved
Answer \( 46 \ \text{mph} \approx 46\ \text{mph} \)
✓ 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 to find the rate of change of the distance.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly sets up the related rates problem using the Pythagorean theorem, differentiates implicitly, and substitutes the given values to find the rate of change of the distance.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the Pythagorean theorem and implicit differentiation to the related rates problem. The substitution of values and final calculation are correct.gpt-oss:20b: pass 2026-09-28
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 2026-09-28 with SymPy 1.14.0.