Newton's law of cooling
Problem 6.352 · hard
An object at 102° is placed in a room at 22°. After 5 minutes it has cooled to 62°. Using Newton's law of cooling, find its temperature after 10 minutes and when it reaches 27°.
- T(t) = Tₐ + (T₀ − Tₐ)e^(−kt) solves dT/dt = −k(T − Tₐ).Reviewed
- \[ \frac{d}{d t} \left(22 + 80 e^{- \frac{t \ln{\left(2 \right)}}{5}}\right) = - 16 e^{- \frac{t \ln{\left(2 \right)}}{5}} \ln{\left(2 \right)} \]The model satisfies the cooling law.✓ Proved
- \[ 62 \]k = ln((T₀ − Tₐ)/(T₁ − Tₐ))/t₁ = log(2)/5 matches the reading at t = 5.✓ Proved
- \[ 42 \]T(10).✓ Proved
- \[ \frac{5 \ln{\left(16 \right)}}{\ln{\left(2 \right)}} = 20 \]Solve T(t) = target: e^(−kt) = (target − Tₐ)/(T₀ − Tₐ), so t = ln((T₀ − Tₐ)/(target − Tₐ))/k.✓ Proved
Answer \( T(10) = 42 \approx 42.00^\circ,\quad t = 20 \approx 20.00\text{ min} \)
✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly applies Newton's Law of Cooling, derives the constant k accurately from the given data points, and computes the requested values for T(10) and the time to reach 27° without error.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies Newton's Law of Cooling, derives the constant k accurately from the given data points, and computes the requested values for T(10) and the time to reach 27° without error.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies Newton's Law of Cooling, derives the constant k accurately from the given data points, and computes the requested values for T(10) and the time to reach 27° without error.
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 2026-10-10 with SymPy 1.14.0.