Related rates
Problem 3.95 · 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 6 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
- \[ \left. 4 \pi r \right|_{\substack{ r=6 }} = 24 \pi \]At r = 6 with dr/dt = 2.✓ Proved
Answer \( 24 \pi \ \text{ft²/s} \approx 75.4\ \text{ft²/s} \)
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 fails to explicitly apply the chain rule to relate dA/dt to dr/dt. Line 2 shows the derivative with respect to r, but the comment incorrectly equates this to dA/dt without multiplying by dr/dt in the equation itself. Line 3 substitutes values into an expression that lacks the dr/dt term, making the derivation logically incomplete and confusing.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly apply the chain rule to relate dA/dt to dr/dt. Line 2 shows the derivative with respect to r, but the comment incorrectly equates this to dA/dt without multiplying by dr/dt in the equation itself. Line 3 substitutes values into an expression that lacks the dr/dt term, making the derivation logically incomplete and confusing.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution incorrectly computes the derivative of the area with respect to radius (dA/dr) instead of time (dA/dt), failing to apply the chain rule properly. It also substitutes r=6 into the expression for dA/dr rather than evaluating dA/dt using both r and dr/dt.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution incorrectly computes the derivative of A with respect to r instead of t, and fails to explicitly apply the chain rule (dA/dt = dA/dr * dr/dt). While the final numerical substitution happens to be correct, the intermediate step 'dA/dt = 2πr dr/dt' is presented as a direct derivative result rather than a consequence of the chain rule, and the substitution line is syntactically confusing.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.