Related rates
Problem 3.262 · medium
Air is pumped into a spherical balloon at 12 cm³/s. How fast is the radius increasing when the radius is 8 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
- \[ \left. \frac{3}{\pi r^{2}} \right|_{\substack{ r=8 }} = \frac{3}{64 \pi} \]dr/dt = (dV/dt)/(4πr²) with r = 8.✓ Proved
Answer \( \frac{3}{64 \pi} \ \text{cm/s} \approx 0.01492\ \text{cm/s} \)
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 (misleading) — The solution omits the given rate dV/dt = 12 cm³/s. It substitutes 3/(πr²) for dV/dt, which is only correct after dividing 12 by 4πr². A student could incorrectly believe 3/(πr²) is the actual rate of change of volume.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly substitute the given rate dV/dt = 12 into the derivative equation before solving for dr/dt. Line 3 substitutes r=8 into an expression that implicitly assumes dV/dt was already divided out, but the algebraic step showing 12 / (4*pi*8^2) is missing, making the derivation incomplete and potentially confusing regarding where the numerator '3' comes from (12/4).
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly substitute the given rate dV/dt = 12 into the derivative equation before solving for dr/dt. Line 3 substitutes r=8 into an expression that implicitly assumes dV/dt was already divided out, but the algebraic step showing 12 / (4*pi*8^2) is missing, making the derivation incomplete and potentially confusing regarding where the numerator '3' comes from (12/4).gpt-oss:20b: fail (misleading) 2026-09-28 — The solution omits the given rate dV/dt = 12 cm³/s. It substitutes 3/(πr²) for dV/dt, which is only correct after dividing 12 by 4πr². A student could incorrectly believe 3/(πr²) is the actual rate of change of volume.qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to substitute the given rate dV/dt = 12 into the derived formula. It implicitly uses dV/dt = 3, resulting in an answer that is 4 times too small.gpt-oss:20b: fail (error) 2026-09-28 — The solution substitutes dV/dt = 3/(πr²) instead of the given rate 12 cm³/s, leading to an incorrect dr/dt. The correct calculation is dr/dt = 12/(4π·8²) = 3/(64π).
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.