Related rates
Problem 3.260 · medium
Air is pumped into a spherical balloon at 16 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
- \[ \left. \frac{4}{\pi r^{2}} \right|_{\substack{ r=8 }} = \frac{1}{16 \pi} \]dr/dt = (dV/dt)/(4πr²) with r = 8.✓ Proved
Answer \( \frac{1}{16 \pi} \ \text{cm/s} \approx 0.01989\ \text{cm/s} \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in 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 dV/dt = 16 cm³/s and incorrectly states dr/dt = 4/(πr²); while the numerical result happens to match, the reasoning is incomplete and could mislead a student about the correct formula.qwen3.6:27b-mlx: fail (error) — The solution fails to substitute the given rate dV/dt = 16 into the equation. It incorrectly uses dV/dt = 1 (implied by the numerator 4 in line 3 being 4*pi*r^2 / 16? No, line 3 shows Subs(4/(pi*r**2)... which implies the formula used was 4/(pi*r^2) or similar, but the correct formula is 16/(4*pi*r^2) = 4/(pi*r^2). Wait, 16/(4*pi*8^2) = 16/(256*pi) = 1/(16*pi). The algebra in line 3 is actually correct for the final answer, but the sentence/explanation is missing the substitution of dV/dt=16. Let's look closer. Line 3 says 'Subs(4/(pi*r**2), r, 8)'. Where did 4/(pi*r^2) come from? dV/dt = 4*pi*r^2 * dr/dt => dr/dt = dV/dt / (4*pi*r^2). If dV/dt = 16, then dr/dt = 16 / (4*pi*r^2) = 4 / (pi*r^2). So the expression 4/(pi*r^2) is correct. The step is technically correct algebraically. However, the sentence in line 3 says 'dr/dt = (dV/dt)/(4πr²) with r = 8'. It does not explicitly state that dV/dt = 16 was substituted to get the numerator 4. A student might think dV/dt is 1 or that the 4 comes from somewhere else. But is it an error? The prompt asks to check if the setup models the problem. The setup is correct. The algebra is correct. The explanation is slightly terse but not false. Let's re-read the strictness. 'nothing would teach a student something false'. If a student sees 4/(pi*r^2) and doesn't see where the 16 went, they might be confused, but it's not false. However, usually, these adversarial reviews look for missing steps. Line 2 defines the derivative relationship. Line 3 jumps to the specific function of r. It skips the explicit substitution of dV/dt = 16. Is this an error? It's a missing step in the logic flow. The sentence says 'dr/dt = (dV/dt)/(4πr²) with r = 8'. It does NOT say 'and dV/dt = 16'. Therefore, the conclusion that the expression is 4/(pi*r^2) does not follow from the sentence provided in line 3 alone, because the sentence doesn't state dV/dt=16. The equation part is marked 'proved', so the algebra is trusted. But the sentence is the part I review. The sentence claims the result follows from 'dr/dt = (dV/dt)/(4πr²) with r = 8'. This is insufficient to derive 4/(pi*r^2) without knowing dV/dt. Thus, the reasoning is incomplete/flawed in the text.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to substitute the given rate dV/dt = 16 into the equation. It incorrectly uses dV/dt = 1 (implied by the numerator 4 in line 3 being 4*pi*r^2 / 16? No, line 3 shows Subs(4/(pi*r**2)... which implies the formula used was 4/(pi*r^2) or similar, but the correct formula is 16/(4*pi*r^2) = 4/(pi*r^2). Wait, 16/(4*pi*8^2) = 16/(256*pi) = 1/(16*pi). The algebra in line 3 is actually correct for the final answer, but the sentence/explanation is missing the substitution of dV/dt=16. Let's look closer. Line 3 says 'Subs(4/(pi*r**2), r, 8)'. Where did 4/(pi*r^2) come from? dV/dt = 4*pi*r^2 * dr/dt => dr/dt = dV/dt / (4*pi*r^2). If dV/dt = 16, then dr/dt = 16 / (4*pi*r^2) = 4 / (pi*r^2). So the expression 4/(pi*r^2) is correct. The step is technically correct algebraically. However, the sentence in line 3 says 'dr/dt = (dV/dt)/(4πr²) with r = 8'. It does not explicitly state that dV/dt = 16 was substituted to get the numerator 4. A student might think dV/dt is 1 or that the 4 comes from somewhere else. But is it an error? The prompt asks to check if the setup models the problem. The setup is correct. The algebra is correct. The explanation is slightly terse but not false. Let's re-read the strictness. 'nothing would teach a student something false'. If a student sees 4/(pi*r^2) and doesn't see where the 16 went, they might be confused, but it's not false. However, usually, these adversarial reviews look for missing steps. Line 2 defines the derivative relationship. Line 3 jumps to the specific function of r. It skips the explicit substitution of dV/dt = 16. Is this an error? It's a missing step in the logic flow. The sentence says 'dr/dt = (dV/dt)/(4πr²) with r = 8'. It does NOT say 'and dV/dt = 16'. Therefore, the conclusion that the expression is 4/(pi*r^2) does not follow from the sentence provided in line 3 alone, because the sentence doesn't state dV/dt=16. The equation part is marked 'proved', so the algebra is trusted. But the sentence is the part I review. The sentence claims the result follows from 'dr/dt = (dV/dt)/(4πr²) with r = 8'. This is insufficient to derive 4/(pi*r^2) without knowing dV/dt. Thus, the reasoning is incomplete/flawed in the text.gpt-oss:20b: fail (misleading) 2026-09-28 — The solution omits the given rate dV/dt = 16 cm³/s and incorrectly states dr/dt = 4/(πr²); while the numerical result happens to match, the reasoning is incomplete and could mislead a student about the correct formula.qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to substitute the given rate dV/dt = 16 into the equation before solving for dr/dt. The final line incorrectly substitutes r=8 into the expression 4/(pi*r^2) instead of 16/(4*pi*r^2), resulting in an answer that is off by a factor of 4.gpt-oss:20b: pass 2026-09-28
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.