Related rates from an equation
Problem 3.519 · easy
\( \displaystyle x \) and \( \displaystyle y \) are functions of \( \displaystyle t \) related by \( \displaystyle x y = 10 \). If \( \displaystyle \frac{dx}{dt} = -2 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = 2 \) and \( \displaystyle y = 5 \).
- \[ 10 \]The point satisfies the equation.✓ Proved
- Differentiate with respect to t: (y)·dx/dt + (x)·dy/dt = 0.
- \[ 5 \]Solve for dy/dt and substitute.✓ Proved
Answer \( \frac{dy}{dt} = 5 \)
Lines: 2 proved, 1 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 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 | moved x along at dx/dt, solved for y numerically, differenced in t |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution claims to solve for dy/dt, but the final equation '5 = 5' is a tautology that does not isolate dy/dt or provide its value. The correct calculation yields dy/dt = 5, but the presented step is mathematically invalid as a solution step.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution claims to solve for dy/dt, but the final equation '5 = 5' is a tautology that does not isolate dy/dt or provide its value. The correct calculation yields dy/dt = 5, but the presented step is mathematically invalid as a solution step.gpt-oss:20b: pass 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution claims to solve for dy/dt and substitute, but the final equation '5 = 5' does not show the calculation or the result. Substituting x=2, y=5, and dx/dt=-2 into 5(-2) + 2(dy/dt) = 0 yields -10 + 2(dy/dt) = 0, so dy/dt = 5. The step is logically incomplete and the final answer is not explicitly derived in the text provided.
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_equation, checked 2026-10-08 with SymPy 1.14.0.