Related rates
Problem 3.94 · medium
A stone dropped in a pond makes a circular ripple whose radius grows at 4 ft/s. How fast is the area inside the ripple growing when the radius is 4 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
- \[ \left. 8 \pi r \right|_{\substack{ r=4 }} = 32 \pi \]At r = 4 with dr/dt = 4.✓ Proved
Answer \( 32 \pi \ \text{ft²/s} \approx 100.5\ \text{ft²/s} \)
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 differentiates with respect to r instead of t, failing to apply the chain rule (dA/dt = 2πr dr/dt). It also implicitly substitutes dr/dt into the derivative expression without stating the relationship, leading to a numerically correct but conceptually flawed derivation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution incorrectly differentiates with respect to r instead of t, failing to apply the chain rule (dA/dt = 2πr dr/dt). It also implicitly substitutes dr/dt into the derivative expression without stating the relationship, leading to a numerically correct but conceptually flawed derivation.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly apply the chain rule to relate dA/dt to dr/dt, instead presenting the derivative with respect to r and then substituting values in a way that obscures the time dependency. While the final numerical answer is correct, the logical step connecting the rate of change of area to the rate of change of radius is missing or incorrectly labeled.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution incorrectly computes the derivative with respect to r instead of t, and the substitution step is algebraically inconsistent with the previous line (substituting r=4 into 8*pi*r implies dr/dt was already substituted, but the derivative step did not show the chain rule application clearly). Specifically, line 2 claims dA/dr = 2*pi*r but labels it dA/dt, which is a fundamental conceptual error in related rates.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.