Related rates from an equation
Problem 3.516 · easy
\( \displaystyle x \) and \( \displaystyle y \) are functions of \( \displaystyle t \) related by \( \displaystyle x^{2} + y^{2} = 169 \). If \( \displaystyle \frac{dx}{dt} = -3 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = 12 \) and \( \displaystyle y = 5 \).
- \[ 169 \]The point satisfies the equation.✓ Proved
- Differentiate with respect to t: (2*x)·dx/dt + (2*y)·dy/dt = 0.Reviewed
- \[ \frac{36}{5} \]Solve for dy/dt and substitute.✓ Proved
Answer \( \frac{dy}{dt} = \frac{36}{5} \)
Lines: 2 proved, 1 reviewed. 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 | 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 |
| 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: pass — The solution correctly applies implicit differentiation and substitutes the given values to find the rate of change. The logic is sound and the final answer is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies implicit differentiation and substitutes the given values to find the rate of change. The logic is sound and the final answer is correct.gpt-oss:20b: pass 2026-10-08gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to substitute the given values (x=12, dx/dt=-3, y=5) into the differentiated equation. It jumps directly to the final answer without showing the calculation -72 + 10(dy/dt) = 0, which is necessary to derive dy/dt = 36/5.
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.