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Newton's law of cooling

Problem 6.318 · 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 30°.

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
\[ \frac{5 \ln{\left(12 \right)}}{\ln{\left(\frac{4}{3} \right)}} = \ln{\left(12^{\frac{5}{\ln{\left(\frac{4}{3} \right)}}} \right)} \]
Solve T(t) = target: e^(−kt) = (target − Tₐ)/(T₀ − Tₐ), so t = ln((T₀ − Tₐ)/(target − Tₐ))/k.✓ Proved

Answer \( T(45) = \frac{1933645}{65536} \approx 29.51^\circ,\quad t = \ln{\left(12^{\frac{5}{\ln{\left(\frac{4}{3} \right)}}} \right)} \approx 43.19\text{ min} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the value of the ambient temperature T_a (25°) and the initial temperature T_0 (85°) in the setup, making the derivation of k and the final formulas opaque. Furthermore, the stated answer for t is mathematically equivalent to the correct value but presented in a confusing, non-simplified form that obscures the standard solution t = (5/k)ln(12).

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to explicitly state the value of the ambient temperature T_a (25°) and the initial temperature T_0 (85°) in the setup, making the derivation of k and the final formulas opaque. Furthermore, the stated answer for t is mathematically equivalent to the correct value but presented in a confusing, non-simplified form that obscures the standard solution t = (5/k)ln(12).
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to explicitly state the derived value of k or the specific function T(t) used for the calculations, jumping from the general formula to unchecked numerical identities. This obscures the critical step of determining the cooling constant, making the solution incomplete and difficult to verify.
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.