Related rates
Problem 3.340 · medium
A stone dropped in a pond makes a circular ripple whose radius grows at 3 ft/s. How fast is the area inside the ripple growing when the radius is 9 ft?
- The area of a circle is A = πr².
- \[ \frac{d}{d r} \pi r^{2} = 2 \pi r \]dA/dt = 2πr dr/dt.✓ Proved
- \[ \left. 6 \pi r \right|_{\substack{ r=9 }} = 54 \pi \]At r = 9 with dr/dt = 3.✓ Proved
Answer \( 54 \pi \ \text{ft²/s} \approx 169.6\ \text{ft²/s} \)
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 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 | the situation was stepped forward and back by a microsecond and the quantity differenced numerically |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly apply the chain rule (dA/dt = dA/dr * dr/dt) in the text, jumping from the derivative with respect to r to the final substitution. While the algebraic result is correct, the logical step connecting dA/dr to dA/t is missing, which is the core concept of related rates.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to explicitly apply the chain rule (dA/dt = dA/dr * dr/dt) in the text, jumping from the derivative with respect to r to the final substitution. While the algebraic result is correct, the logical step connecting dA/dr to dA/t is missing, which is the core concept of related rates.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution incorrectly computes the derivative of the area with respect to time. It substitutes dr/dt into the derivative with respect to r (2πr) to get 6πr, but fails to explicitly apply the chain rule (dA/dt = dA/dr * dr/dt) or show the intermediate step dA/dr = 2πr. While the numerical substitution is correct, the logical leap from 'Derivative(..., r)' to the final time derivative is mathematically unjustified and skips the crucial application of the chain rule.gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.