Related rates
Problem 3.105 · medium
Car A drives east from an intersection at 40 mph and car B drives north from it at 56 mph. How fast is the distance between them changing when A is 6 mi and B is 8 mi from the intersection?
- With A at distance a and B at distance b, the distance between them satisfies D² = a² + b².
- Differentiate: 2D dD/dt = 2a da/dt + 2b db/dt.
- \[ \left. \sqrt{x^{2} + y^{2}} \right|_{\substack{ x=6\\ y=8 }} = 10 \]D at that moment.✓ Proved
- \[ \left. \frac{40 x + 56 y}{\sqrt{x^{2} + y^{2}}} \right|_{\substack{ x=6\\ y=8 }} = \frac{344}{5} \]dD/dt = (a·da/dt + b·db/dt)/D with a = 6, b = 8.✓ Proved
Answer \( \frac{344}{5} \ \text{mph} \approx 68.8\ \text{mph} \)
Lines: 2 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | Not checked | — | 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: fail (misleading) — The solution omits the crucial step of assigning the given speeds to da/dt and db/dt (40 mph and 56 mph). Without explicitly stating that these are the rates of change of a and b, a student could be misled into thinking the formula for dD/dt was derived without using the given speeds. The rest of the setup and differentiation are correct, but the missing justification makes the solution misleading.qwen3.6:27b-mlx: pass — The solution correctly applies the Pythagorean theorem and implicit differentiation. The substitution of values and final calculation are accurate.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the Pythagorean theorem and implicit differentiation. The substitution of values and final calculation are accurate.gpt-oss:20b: fail (misleading) 2026-09-28 — The solution omits the crucial step of assigning the given speeds to da/dt and db/dt (40 mph and 56 mph). Without explicitly stating that these are the rates of change of a and b, a student could be misled into thinking the formula for dD/dt was derived without using the given speeds. The rest of the setup and differentiation are correct, but the missing justification makes the solution misleading.qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: pass 2026-09-26
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.