Related rates
Problem 3.14 · easy
Air is pumped into a spherical balloon at 8 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
- \[ \left. \frac{2}{\pi r^{2}} \right|_{\substack{ r=8 }} = \frac{1}{32 \pi} \]dr/dt = (dV/dt)/(4πr²) with r = 8.✓ Proved
Answer \( \frac{1}{32 \pi} \ \text{cm/s} \approx 0.009947\ \text{cm/s} \)
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. 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: fail (error) — The solution never uses the given rate dV/dt = 8 cm³/s. Instead it substitutes 2/(π r²), which is incorrect. The correct computation is dr/dt = (8)/(4π·8²) = 1/(32π).qwen3.6:27b-mlx: fail (error) — The solution substitutes dV/dt = 2 cm³/s instead of the given 8 cm³/s. This leads to an incorrect final answer.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution substitutes dV/dt = 2 cm³/s instead of the given 8 cm³/s. This leads to an incorrect final answer.gpt-oss:20b: fail (error) 2026-09-28 — The solution never uses the given rate dV/dt = 8 cm³/s. Instead it substitutes 2/(π r²), which is incorrect. The correct computation is dr/dt = (8)/(4π·8²) = 1/(32π).qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution incorrectly substitutes dV/dt = 2 instead of the given 8 cm³/s. Additionally, the chain rule application is stated as dV/dt = 4πr² dr/dt, but the substitution step implies a different value for dV/dt without justification.gpt-oss:20b: fail (error) 2026-09-27 — The solution never uses the given rate of change of volume (8 cm³/s). In step 3 the numerator 2/(π r²) is incorrect; it should be 8/(4πr²), yielding dr/dt = 8/(4π·8²) = 1/(32π).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 from the derivative with respect to r to a substitution that assumes dV/dt = 2. The problem states dV/dt = 8, but the substitution line uses 2 in the numerator (2/(pi*r**2)), leading to an incorrect final answer of 1/(32π) instead of 1/(64π).gpt-oss:20b: fail (misleading) 2026-09-26 — The solution substitutes an incorrect expression for dV/dt (2/(πr²)) instead of the given rate 8 cm³/s. While the final numerical value happens to be correct, the reasoning is flawed and would mislead a student.
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.