Newton's law of cooling
Problem 6.311 · hard
An object at 85° is placed in a room at 25°. After 5 minutes it has cooled to 70°. Using Newton's law of cooling, find its temperature after 45 minutes and when it reaches 35°.
- T(t) = Tₐ + (T₀ − Tₐ)e^(−kt) solves dT/dt = −k(T − Tₐ).
- \[ \frac{d}{d t} \left(25 + 60 e^{- \frac{t \ln{\left(\frac{4}{3} \right)}}{5}}\right) = - 12 e^{- \frac{t \ln{\left(\frac{4}{3} \right)}}{5}} \ln{\left(\frac{4}{3} \right)} \]The model satisfies the cooling law.✓ Proved
- \[ 70 \]k = ln((T₀ − Tₐ)/(T₁ − Tₐ))/t₁ = log(4/3)/5 matches the reading at t = 5.✓ Proved
- \[ \frac{1933645}{65536} \]T(45).✓ Proved
- \[ \frac{5 \ln{\left(6 \right)}}{\ln{\left(\frac{4}{3} \right)}} = \ln{\left(6^{\frac{5}{\ln{\left(\frac{4}{3} \right)}}} \right)} \]Solve T(t) = target: e^(−kt) = (target − Tₐ)/(T₀ − Tₐ), so t = ln((T₀ − Tₐ)/(target − Tₐ))/k.✓ Proved
Answer \( T(45) = \frac{1933645}{65536} \approx 29.51^\circ,\quad t = \ln{\left(6^{\frac{5}{\ln{\left(\frac{4}{3} \right)}}} \right)} \approx 31.14\text{ min} \)
Lines: 4 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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ 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 | k fitted by a root-finder, then the cooling law integrated numerically |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution fails to answer the specific question asked: it calculates the time to reach 35°C (approx 31.14 min) but does not provide the temperature at 45 minutes in the final stated answer, despite calculating it in step 4. Furthermore, the final answer for time is presented in a confusing, non-simplified logarithmic form rather than a numerical value consistent with the temperature approximation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to answer the specific question asked: it calculates the time to reach 35°C (approx 31.14 min) but does not provide the temperature at 45 minutes in the final stated answer, despite calculating it in step 4. Furthermore, the final answer for time is presented in a confusing, non-simplified logarithmic form rather than a numerical value consistent with the temperature approximation.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution calculates the time to reach 35° as approximately 31.14 minutes, which is less than the 45 minutes used for the first part of the question. Since the object is cooling from 85° to 25°, it must take longer to reach 35° (closer to ambient) than to reach ~29.5°. The formula for t in line 5 is correct, but the numerical evaluation in the stated answer is inconsistent with the physics (T(31.14) would be higher than T(45), so T(31.14) cannot be 35 if T(45) is 29.5). Specifically, T(45) ≈ 29.51, so T(31.14) must be > 29.51. Let's check T(31.14). k = ln(4/3)/5 ≈ 0.0559. t=31.14. kt ≈ 1.74. e^-1.74 ≈ 0.175. T = 25 + 60*0.75^9? No. T = 25 + 60 * exp(-k*31.14). 45 min gives T=29.51. 31.14 min is earlier, so T should be higher. 35 is higher than 29.51. So t for 35 should be less than 45. Wait. 85 -> 70 (5 min). 70 -> 29.5 (45 min). 35 is between 70 and 29.5. So t for 35 should be between 5 and 45. 31.14 is between 5 and 45. Is it correct? Let's re-calculate. k = ln(4/3)/5. t = ln(60/10)/k = ln(6)/k = 5*ln(6)/ln(4/3). ln(6) ≈ 1.7917. ln(4/3) ≈ 0.2877. t ≈ 5 * 1.7917 / 0.2877 ≈ 31.14. The calculation is actually correct. Why did I think it was wrong? T(45) = 25 + 60 * exp(-45 * ln(4/3)/5) = 25 + 60 * (3/4)^9 = 25 + 60 * 19683/262144 ≈ 25 + 4.51 = 29.51. T(31.14) = 25 + 60 * exp(-31.14 * k). 31.14 * k = ln(6). exp(-ln(6)) = 1/6. T = 25 + 60/6 = 35. The math is correct. The verdict should be pass. Let me double check the prompt constraints. 'find its temperature after 45 minutes and when it reaches 35'. The solution provides both. The algebra is checked. The sentences are standard. Is there a subtle error? Line 1: T(t) = Ta + (T0-Ta)e^-kt. Correct. Line 5: t = ln((T0-Ta)/(target-Ta))/k. Correct. The stated answer has t = ln(6^(5/ln(4/3))). This is equal to 5/ln(4/3) * ln(6). This matches. The approximation 31.14 is correct. The approximation 29.51 is correct. Everything seems correct.gpt-oss:20b: pass 2026-10-08
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/newtons_cooling, checked 2026-10-08 with SymPy 1.14.0.