Related rates
Problem 3.103 · medium
A stone dropped in a pond makes a circular ripple whose radius grows at 2 ft/s. How fast is the area inside the ripple growing when the radius is 14 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
- \[ \left. 4 \pi r \right|_{\substack{ r=14 }} = 56 \pi \]At r = 14 with dr/dt = 2.✓ Proved
Answer \( 56 \pi \ \text{ft²/s} \approx 175.9\ \text{ft²/s} \)
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. How we verify
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 incorrectly computes the derivative of the area with respect to radius (dA/dr) instead of time (dA/dt). It fails to apply the chain rule to relate dA/dt to dr/dt, resulting in a missing factor of 2 in the final answer.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution incorrectly computes the derivative of the area with respect to radius (dA/dr) instead of time (dA/dt). It fails to apply the chain rule to relate dA/dt to dr/dt, resulting in a missing factor of 2 in the final answer.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution incorrectly computes the derivative with respect to r instead of t, and then substitutes dr/dt into the resulting expression without showing the chain rule step (dA/dt = 2πr dr/dt). This skips the crucial logical step of relating the rates.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly apply the chain rule to relate dA/dt to dr/dt. Step 2 computes the partial derivative with respect to r, but the text annotation claims this equals dA/dt, which is dimensionally and mathematically incorrect without multiplying by dr/dt. The final substitution in Step 3 implicitly includes dr/dt=2, but the logical gap in Step 2 is a significant error in a calculus context.gpt-oss:20b: pass 2026-09-26
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 2026-09-28 with SymPy 1.14.0.