Related rates
Problem 3.256 · medium
Air is pumped into a spherical balloon at 8 cm³/s. How fast is the radius increasing when the radius is 6 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
- \[ \left. \frac{2}{\pi r^{2}} \right|_{\substack{ r=6 }} = \frac{1}{18 \pi} \]dr/dt = (dV/dt)/(4πr²) with r = 6.✓ Proved
Answer \( \frac{1}{18 \pi} \ \text{cm/s} \approx 0.01768\ \text{cm/s} \)
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 never uses the given rate dV/dt = 8 cm³/s; instead it substitutes an incorrect expression 2/(πr²). The correct computation is dr/dt = (8)/(4π·6²) = 1/(18π) cm/s.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly substitute the given rate dV/dt = 8 into the derived formula. Line 3 substitutes r=6 into an expression that implicitly assumes dV/dt=2 (since 2/(pi*6^2) = 1/(18pi)), but the problem states dV/dt = 8. The correct answer should be 8/(4*pi*36) = 1/(18pi) only if dV/dt was 2, but with dV/dt=8, the answer is 8/(144pi) = 1/(18pi). Wait, let me re-calculate. dV/dt = 8. 4*pi*r^2 = 4*pi*36 = 144*pi. dr/dt = 8 / 144*pi = 1 / 18*pi. The numerical answer is actually correct. However, the intermediate step in line 3 shows `Subs(2/(pi*r**2), r, 6)`. Where did the 2 come from? The formula is dr/dt = (dV/dt) / (4*pi*r^2). If dV/dt = 8, then dr/dt = 8 / (4*pi*r^2) = 2 / (pi*r^2). So the expression `2/(pi*r**2)` is correct for this specific problem. The sentence in line 3 says `dr/dt = (dV/dt)/(4πr²) with r = 6`. It does not explicitly state that dV/dt=8 was used to simplify the coefficient to 2. While the math is correct, the transition from the general formula to the specific substitution is opaque. A student might wonder where the 2 came from. Is it misleading? It skips the substitution of dV/dt=8. It's a bit of a jump. Let's look closer. Line 2 establishes dV/dt = 4*pi*r^2 * dr/dt. Line 3 substitutes into `2/(pi*r**2)`. This expression `2/(pi*r**2)` is equal to `8/(4*pi*r**2)`. So the solver simplified `8/4` to `2` before substituting r. This is valid algebra. The text description `dr/dt = (dV/dt)/(4πr²) with r = 6` is slightly incomplete because it doesn't mention substituting dV/dt=8, but the equation shown `Subs(2/(pi*r**2)...)` reflects the correct numerical value. Is it an error? No. Is it misleading? Potentially, if a student doesn't realize the 2 comes from 8/4. But it's not false. It's just a compressed step. However, usually, we want to see the substitution of all knowns. Let's check the verdict criteria. 'Nothing would teach a student something false.' The student might think the formula for dr/dt is always 2/(pi*r^2) if they don't look closely. That is a risk. But the line says `Subs(...)`, implying a specific instance. I will call it 'misleading' because it obscures the dependency on dV/dt=8.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly substitute the given rate dV/dt = 8 into the derived formula. Line 3 substitutes r=6 into an expression that implicitly assumes dV/dt=2 (since 2/(pi*6^2) = 1/(18pi)), but the problem states dV/dt = 8. The correct answer should be 8/(4*pi*36) = 1/(18pi) only if dV/dt was 2, but with dV/dt=8, the answer is 8/(144pi) = 1/(18pi). Wait, let me re-calculate. dV/dt = 8. 4*pi*r^2 = 4*pi*36 = 144*pi. dr/dt = 8 / 144*pi = 1 / 18*pi. The numerical answer is actually correct. However, the intermediate step in line 3 shows `Subs(2/(pi*r**2), r, 6)`. Where did the 2 come from? The formula is dr/dt = (dV/dt) / (4*pi*r^2). If dV/dt = 8, then dr/dt = 8 / (4*pi*r^2) = 2 / (pi*r^2). So the expression `2/(pi*r**2)` is correct for this specific problem. The sentence in line 3 says `dr/dt = (dV/dt)/(4πr²) with r = 6`. It does not explicitly state that dV/dt=8 was used to simplify the coefficient to 2. While the math is correct, the transition from the general formula to the specific substitution is opaque. A student might wonder where the 2 came from. Is it misleading? It skips the substitution of dV/dt=8. It's a bit of a jump. Let's look closer. Line 2 establishes dV/dt = 4*pi*r^2 * dr/dt. Line 3 substitutes into `2/(pi*r**2)`. This expression `2/(pi*r**2)` is equal to `8/(4*pi*r**2)`. So the solver simplified `8/4` to `2` before substituting r. This is valid algebra. The text description `dr/dt = (dV/dt)/(4πr²) with r = 6` is slightly incomplete because it doesn't mention substituting dV/dt=8, but the equation shown `Subs(2/(pi*r**2)...)` reflects the correct numerical value. Is it an error? No. Is it misleading? Potentially, if a student doesn't realize the 2 comes from 8/4. But it's not false. It's just a compressed step. However, usually, we want to see the substitution of all knowns. Let's check the verdict criteria. 'Nothing would teach a student something false.' The student might think the formula for dr/dt is always 2/(pi*r^2) if they don't look closely. That is a risk. But the line says `Subs(...)`, implying a specific instance. I will call it 'misleading' because it obscures the dependency on dV/dt=8.gpt-oss:20b: fail (error) 2026-09-28 — The solution never uses the given rate dV/dt = 8 cm³/s; instead it substitutes an incorrect expression 2/(πr²). The correct computation is dr/dt = (8)/(4π·6²) = 1/(18π) cm/s.qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution incorrectly substitutes dV/dt = 2 instead of the given 8 cm³/s, leading to an incorrect final answer.gpt-oss:20b: fail (error) 2026-09-28 — The solution never uses the given rate of change of volume (8 cm³/s). In step 3 the expression 2/(π r²) is incorrect; it should be 8/(4πr²) evaluated at r=6, yielding dr/dt=8/(4π·36)=1/(18π). The omitted substitution makes the conclusion invalid.
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.