Related rates
Problem 3.1 · easy
Air is pumped into a spherical balloon at 4 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
- \[ \left. 1 \frac{1}{\pi r^{2}} \right|_{\substack{ r=7 }} = \frac{1}{49 \pi} \]dr/dt = (dV/dt)/(4πr²) with r = 7.✓ Proved
Answer \( \frac{1}{49 \pi} \ \text{cm/s} \approx 0.006496\ \text{cm/s} \)
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 of change of volume (4 cm³/s). In step 3 the expression 1/(πr²) is inserted for dV/dt, which is incorrect. The correct computation should use dV/dt = 4 and then dr/dt = 4/(4πr²)=1/(πr²), yielding dr/dt = 1/(49π) when r = 7. The omission of the given rate makes the reasoning invalid.qwen3.6:27b-mlx: fail (error) — The solution fails to substitute the given rate dV/dt = 4 cm³/s into the equation. It incorrectly assumes dV/dt = 1, leading to the wrong answer 1/(49π) instead of the correct 1/(49π/3) = 3/(196π).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to substitute the given rate dV/dt = 4 cm³/s into the equation. It incorrectly assumes dV/dt = 1, leading to the wrong answer 1/(49π) instead of the correct 1/(49π/3) = 3/(196π).gpt-oss:20b: fail (error) 2026-09-28 — The solution never uses the given rate of change of volume (4 cm³/s). In step 3 the expression 1/(πr²) is inserted for dV/dt, which is incorrect. The correct computation should use dV/dt = 4 and then dr/dt = 4/(4πr²)=1/(πr²), yielding dr/dt = 1/(49π) when r = 7. The omission of the given rate makes the reasoning invalid.qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly apply the chain rule to relate dV/dt and dr/dt, jumping from the derivative with respect to r to the final answer without showing the division by dV/dt or the substitution of dV/dt = 4. While the final numerical answer is correct, the logical steps are missing, making it impossible to verify the method was applied correctly.gpt-oss:20b: fail (error) 2026-09-27 — The solution incorrectly substitutes dV/dt as 1/(πr²) instead of the given 4 cm³/s, leading to an incorrect dr/dt value.qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to substitute the given rate dV/dt = 4 cm³/s into the equation, resulting in a final answer that is off by a factor of 4. The step labeled 'Subs' incorrectly assumes dV/dt = 1 or ignores the given value entirely.gpt-oss:20b: fail (misleading) 2026-09-26 — The solution omits the given rate dV/dt = 4 cm³/s when computing dr/dt, which could mislead a student into thinking the rate is 1 instead of 4. The final numerical result is correct, but the missing step makes the reasoning incomplete.
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.