Related rates
Problem 3.299 · medium
Air is pumped into a spherical balloon at 4 cm³/s. How fast is the radius increasing when the radius is 10 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
- \[ \left. 1 \frac{1}{\pi r^{2}} \right|_{\substack{ r=10 }} = \frac{1}{100 \pi} \]dr/dt = (dV/dt)/(4πr²) with r = 10.✓ Proved
Answer \( \frac{1}{100 \pi} \ \text{cm/s} \approx 0.003183\ \text{cm/s} \)
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 substitute the given rate dV/dt = 4 cm³/s into the equation, resulting in a final answer that is off by a factor of 4. The step labeled 'proved' computes 1/(πr²) instead of 4/(4πr²) = 1/(πr²) * (dV/dt)/4, effectively assuming dV/dt = 1.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to substitute the given rate dV/dt = 4 cm³/s into the equation, resulting in a final answer that is off by a factor of 4. The step labeled 'proved' computes 1/(πr²) instead of 4/(4πr²) = 1/(πr²) * (dV/dt)/4, effectively assuming dV/dt = 1.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution fails to explicitly substitute the given rate dV/dt = 4 cm³/s into the derived formula. It incorrectly implies dr/dt = 1/(πr²) instead of dr/dt = 4/(4πr²), leading to a result that is off by a factor of 4.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution fails to explicitly substitute the given rate dV/dt = 4 cm³/s into the derived formula. It calculates 1/(4πr²) instead of 4/(4πr²), resulting in an answer that is off by a factor of 4.gpt-oss:20b: fail (error) 2026-09-29 — The solution omits the given rate of change of volume (4 cm³/s). In step 3 it incorrectly substitutes 1/(πr²) for dV/dt, leading to an incorrect dr/dt. The correct computation should use dr/dt = (4)/(4πr²) = 1/(πr²) and then evaluate at r=10 to get 1/(100π).
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.