Related rates
Problem 3.219 · medium
Air is pumped into a spherical balloon at 16 cm³/s. How fast is the radius increasing when the radius is 7 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
- \[ \left. \frac{4}{\pi r^{2}} \right|_{\substack{ r=7 }} = \frac{4}{49 \pi} \]dr/dt = (dV/dt)/(4πr²) with r = 7.✓ Proved
Answer \( \frac{4}{49 \pi} \ \text{cm/s} \approx 0.02598\ \text{cm/s} \)
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 omits the given rate dV/dt = 16 cm³/s. In step 3 the expression 4/(πr²) assumes dV/dt = 4, which is incorrect. The correct calculation is dr/dt = 16/(4π·7²) = 4/(49π).qwen3.6:27b-mlx: fail (error) — The solution fails to substitute the given rate dV/dt = 16 into the equation. It incorrectly uses a value of 4 for dV/dt (likely confusing the coefficient 4/3 with the rate) to arrive at the final answer, resulting in an incorrect numerical result.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to substitute the given rate dV/dt = 16 into the equation. It incorrectly uses a value of 4 for dV/dt (likely confusing the coefficient 4/3 with the rate) to arrive at the final answer, resulting in an incorrect numerical result.gpt-oss:20b: fail (error) 2026-09-27 — The solution omits the given rate dV/dt = 16 cm³/s. In step 3 the expression 4/(πr²) assumes dV/dt = 4, which is incorrect. The correct calculation is dr/dt = 16/(4π·7²) = 4/(49π).qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to substitute the given rate dV/dt = 16 into the equation. It incorrectly uses a numerator of 4 in the final substitution step, leading to an answer that is off by a factor of 4.gpt-oss:20b: fail (error) 2026-09-27 — The solution never uses the given rate of air inflow (16 cm³/s). In step 3 the expression 4/(πr²) is derived from 16/(4πr²), but the 16 is omitted, leading to an incorrect value for dr/dt.
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 2026-09-27 with SymPy 1.14.0.