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Problem 3.13 · easy
Car A drives east from an intersection at 59 mph and car B drives north from it at 33 mph. How fast is the distance between them changing when A is 3 mi and B is 4 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.
- \[ 5 \]D at that moment.✓ Proved
- \[ \left. \frac{59 x + 33 y}{\sqrt{x^{2} + y^{2}}} \right|_{\substack{ x=3\\ y=4 }} = \frac{309}{5} \]dD/dt = (a·da/dt + b·db/dt)/D with a = 3, b = 4.✓ Proved
Answer \( \frac{309}{5} \ \text{mph} \approx 61.8\ \text{mph} \)
Lines: 2 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
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 |
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 2026-09-26 with SymPy 1.14.0.