Related rates
Problem 3.296 · medium
Air is pumped into a spherical balloon at 100 cm³/s. How fast is the radius increasing when the radius is 2 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
- \[ \left. \frac{25}{\pi r^{2}} \right|_{\substack{ r=2 }} = \frac{25}{4 \pi} \]dr/dt = (dV/dt)/(4πr²) with r = 2.✓ Proved
Answer \( \frac{25}{4 \pi} \ \text{cm/s} \approx 1.989\ \text{cm/s} \)
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 (misleading) — The solution omits the crucial step of substituting the given rate dV/dt = 100 cm³/s into the formula dr/dt = (dV/dt)/(4πr²). Without that sentence a student might not see where the 25 comes from, leading to confusion.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly substitute the given rate dV/dt = 100 cm³/s into the equation. It jumps from the derivative formula to a substitution of 25/(pi*r**2), implying dV/dt was 100 but skipping the algebraic step of dividing 100 by 4 to get 25, which is confusing and technically incomplete reasoning.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to explicitly substitute the given rate dV/dt = 100 cm³/s into the equation. It jumps from the derivative formula to a substitution of 25/(pi*r**2), implying dV/dt was 100 but skipping the algebraic step of dividing 100 by 4 to get 25, which is confusing and technically incomplete reasoning.gpt-oss:20b: fail (misleading) 2026-10-03 — The solution omits the crucial step of substituting the given rate dV/dt = 100 cm³/s into the formula dr/dt = (dV/dt)/(4πr²). Without that sentence a student might not see where the 25 comes from, leading to confusion.qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution fails to explicitly state the chain rule relationship dV/dt = 4πr² dr/dt, jumping directly to a substitution that implies dr/dt = 25/(πr²) without showing how the given rate of 100 cm³/s was used to derive that expression. Specifically, step 3 substitutes 25/(πr²) for dr/dt, but 100 / (4πr²) simplifies to 25/(πr²), so the algebra is correct, but the logical step of dividing dV/dt by 4πr² is missing from the text, making the derivation opaque and potentially confusing for a student.gpt-oss:20b: fail (misleading) 2026-09-29 — The solution omits the key step of substituting the given rate dV/dt = 100 cm³/s. It uses 25/(πr²) without explaining that this equals 100/(4πr²), so a student could be misled about how the numerical value was obtained.qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution fails to explicitly substitute the given rate dV/dt = 100 cm³/s into the derived equation. It jumps from the general derivative to a specific numerical answer without showing the intermediate step dr/dt = 100 / (4πr²), making the logic discontinuous and the source of the '25' unclear.gpt-oss:20b: fail (error) 2026-09-29 — The solution incorrectly uses 25 instead of the given rate 100 cm³/s, 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 2026-10-03 with SymPy 1.14.0.