Related rates
Problem 3.301 · medium
Air is pumped into a spherical balloon at 8 cm³/s. How fast is the radius increasing when the radius is 4 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
- \[ \left. \frac{2}{\pi r^{2}} \right|_{\substack{ r=4 }} = \frac{1}{8 \pi} \]dr/dt = (dV/dt)/(4πr²) with r = 4.✓ Proved
Answer \( \frac{1}{8 \pi} \ \text{cm/s} \approx 0.03979\ \text{cm/s} \)
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 of 8 cm³/s and incorrectly substitutes 2/(πr²) for dV/dt, which misleads a student about how to apply the given information.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly substitute the given rate dV/dt = 8 cm³/s into the derived formula. Line 3 substitutes r=4 into an expression that implicitly assumes dV/dt=2, leading to an incorrect numerical result (1/(8π) instead of 1/(16π)).
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution fails to explicitly substitute the given rate dV/dt = 8 cm³/s into the derived formula. Line 3 substitutes r=4 into an expression that implicitly assumes dV/dt=2, leading to an incorrect numerical result (1/(8π) instead of 1/(16π)).gpt-oss:20b: fail (misleading) 2026-09-29 — The solution omits the given rate of 8 cm³/s and incorrectly substitutes 2/(πr²) for dV/dt, which misleads a student about how to apply the given information.qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution fails to explicitly substitute the given rate dV/dt = 8 into the derived formula before solving for dr/dt. It jumps from the general relationship to a specific numerical substitution without showing the intermediate step of dividing 8 by 4πr², which obscures the logic and risks calculation errors.gpt-oss:20b: fail (error) 2026-09-29 — The third line substitutes an incorrect expression for dV/dt (2/(π r²) instead of the given 8 cm³/s), yet still arrives at the correct numerical answer. This misrepresents the data and could 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 2026-09-29 with SymPy 1.14.0.