Related rates from an equation
Problem 3.619 · easy
\( \displaystyle x \) and \( \displaystyle y \) are functions of \( \displaystyle t \) related by \( \displaystyle x^{2} + y^{2} = 100 \). If \( \displaystyle \frac{dx}{dt} = -1 \), find \( \displaystyle \frac{dy}{dt} \) when \( \displaystyle x = 6 \) and \( \displaystyle y = 8 \).
- \[ 100 \]The point satisfies the equation.✓ Proved
- Differentiate with respect to t: (2*x)·dx/dt + (2*y)·dy/dt = 0.
- \[ \frac{3}{4} \]Solve for dy/dt and substitute.✓ Proved
Answer \( \frac{dy}{dt} = \frac{3}{4} \)
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 (misleading) — The solution omits the crucial step of substituting the specific values x=6, y=8, and dx/dt=-1 into the differentiated equation. Without showing this substitution, the student cannot see how the signs and magnitudes lead to the result, making the derivation opaque and potentially confusing regarding sign handling.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (misleading) 2026-10-13 — The solution omits the crucial step of substituting the specific values x=6, y=8, and dx/dt=-1 into the differentiated equation. Without showing this substitution, the student cannot see how the signs and magnitudes lead to the result, making the derivation opaque and potentially confusing regarding sign handling.gpt-oss:20b: fail (error) 2026-10-13 — The solution omits the crucial substitution of the given values x=6, y=8, and dx/dt=-1 into the differentiated equation, so it does not actually compute dy/dt. The first line is unnecessary, and the second line is correct but incomplete. The conclusion is unsupported.qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly applies implicit differentiation and substitutes the given values to find the rate of change.
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 2026-10-13 with SymPy 1.14.0.