Related rates
Problem 3.176 · medium
Air is pumped into a spherical balloon at 4 cm³/s. How fast is the radius increasing when the radius is 6 cm?
- The volume of a sphere is V = (4/3)πr³.
- \[ \frac{d}{d r} \frac{4 \pi r^{3}}{3} = 4 \pi r^{2} \]dV/dt = 4πr² dr/dt.✓ Proved
- \[ \left. \frac{1}{\pi r^{2}} \right|_{\substack{ r=6 }} = \frac{1}{36 \pi} \]dr/dt = (dV/dt)/(4πr²) with r = 6.✓ Proved
Answer \( \frac{1}{36 \pi} \ \text{cm/s} \approx 0.008842\ \text{cm/s} \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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: fail (error) — The solution never uses the given rate dV/dt = 4 cm³/s. Instead it substitutes 1/(πr²) for dV/dt, which is incorrect. The correct computation is dr/dt = (4)/(4π·6²) = 1/(36π).qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the chain rule relationship dV/dt = 4πr²(dr/dt), jumping directly to the final substitution. While the algebraic result is correct, the logical step connecting the derivative with respect to r to the rate of change with respect to time is missing or implied without justification, which is a critical conceptual step in related rates problems.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly state the chain rule relationship dV/dt = 4πr²(dr/dt), jumping directly to the final substitution. While the algebraic result is correct, the logical step connecting the derivative with respect to r to the rate of change with respect to time is missing or implied without justification, which is a critical conceptual step in related rates problems.gpt-oss:20b: fail (error) 2026-09-26 — The solution never uses the given rate dV/dt = 4 cm³/s. Instead it substitutes 1/(πr²) for dV/dt, which is incorrect. The correct computation is dr/dt = (4)/(4π·6²) = 1/(36π).qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly substitute the given rate dV/dt = 4 cm³/s into the derived formula. It incorrectly implies that dr/dt = 1/(pi*r^2) generally, omitting the factor of 4 from the volume rate, leading to a result that is off by a factor of 4.gpt-oss:20b: fail (misleading) 2026-09-26 — The solution omits the given rate dV/dt = 4 cm³/s; it substitutes 1/(πr²) directly, which is the result of dividing 4 by 4πr², but this step is not justified in the text. A student could be confused about where the 4 comes from.
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 2026-09-26 with SymPy 1.14.0.