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Newton's law of cooling

Problem 6.235 · hard

An object at 62° is placed in a room at 22°. After 5 minutes it has cooled to 42°. Using Newton's law of cooling, find its temperature after 45 minutes and when it reaches 37°.

T(t) = Tₐ + (T₀ − Tₐ)e^(−kt) solves dT/dt = −k(T − Tₐ).
Reviewed
\[ \frac{d}{d t} \left(22 + 40 e^{- \frac{t \ln{\left(2 \right)}}{5}}\right) = - 8 e^{- \frac{t \ln{\left(2 \right)}}{5}} \ln{\left(2 \right)} \]
The model satisfies the cooling law.✓ Proved
\[ 42 \]
k = ln((T₀ − Tₐ)/(T₁ − Tₐ))/t₁ = log(2)/5 matches the reading at t = 5.✓ Proved
\[ \frac{1413}{64} \]
T(45).✓ Proved
\[ \frac{5 \ln{\left(\frac{8}{3} \right)}}{\ln{\left(2 \right)}} = - \frac{5 \ln{\left(3 \right)}}{\ln{\left(2 \right)}} + 15 \]
Solve T(t) = target: e^(−kt) = (target − Tₐ)/(T₀ − Tₐ), so t = ln((T₀ − Tₐ)/(target − Tₐ))/k.✓ Proved

Answer \( T(45) = \frac{1413}{64} \approx 22.08^\circ,\quad t = - \frac{5 \ln{\left(3 \right)}}{\ln{\left(2 \right)}} + 15 \approx 7.08\text{ min} \)

✓ 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: pass
qwen3.6:27b-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(45) and the time to reach 37°. The algebraic steps are verified and the final expressions match the stated answer.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — 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(45) and the time to reach 37°. The algebraic steps are verified and the final expressions match the stated answer.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies Newton's Law of Cooling, derives the decay constant k = ln(2)/5 from the given data, and accurately computes the temperature at t=45 and the time to reach 37°. The algebraic steps and final numerical approximations are correct.
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.