Newton's law of cooling
Problem 6.276 · hard
An object at 148° is placed in a room at 68°. After 5 minutes it has cooled to 108°. Using Newton's law of cooling, find its temperature after 10 minutes and when it reaches 78°.
- T(t) = Tₐ + (T₀ − Tₐ)e^(−kt) solves dT/dt = −k(T − Tₐ).Reviewed
- \[ \frac{d}{d t} \left(68 + 80 e^{- \frac{t \ln{\left(2 \right)}}{5}}\right) = - 16 e^{- \frac{t \ln{\left(2 \right)}}{5}} \ln{\left(2 \right)} \]The model satisfies the cooling law.✓ Proved
- \[ 108 \]k = ln((T₀ − Tₐ)/(T₁ − Tₐ))/t₁ = log(2)/5 matches the reading at t = 5.✓ Proved
- \[ 88 \]T(10).✓ Proved
- \[ \frac{5 \ln{\left(8 \right)}}{\ln{\left(2 \right)}} = 15 \]Solve T(t) = target: e^(−kt) = (target − Tₐ)/(T₀ − Tₐ), so t = ln((T₀ − Tₐ)/(target − Tₐ))/k.✓ Proved
Answer \( T(10) = 88 \approx 88.00^\circ,\quad t = 15 \approx 15.00\text{ min} \)
✓ 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 from the given data, and accurately computes the requested temperature and time.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies Newton's Law of Cooling, derives the constant k from the given data, and accurately computes the requested temperature and time.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies Newton's Law of Cooling, derives the constant k from the given data points, and accurately computes the temperature at t=10 and the time to reach 78°. The logic is sound and the calculations are correct.
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 2026-10-06 with SymPy 1.14.0.