Home›Calculus 2›Newton's law of cooling›Problem 6.238

Newton's law of cooling

Problem 6.238 · hard

An object at 172° is placed in a room at 22°. After 20 minutes it has cooled to 97°. Using Newton's law of cooling, find its temperature after 45 minutes and when it reaches 32°.

T(t) = Tₐ + (T₀ − Tₐ)e^(−kt) solves dT/dt = −k(T − Tₐ).
Reviewed
\[ \frac{d}{d t} \left(22 + 150 e^{- \frac{t \ln{\left(2 \right)}}{20}}\right) = - \frac{15 e^{- \frac{t \ln{\left(2 \right)}}{20}} \ln{\left(2 \right)}}{2} \]
The model satisfies the cooling law.✓ Proved
\[ 97 \]
k = ln((T₀ − Tₐ)/(T₁ − Tₐ))/t₁ = log(2)/20 matches the reading at t = 20.✓ Proved
\[ 22 + \frac{75 \cdot 2^{\frac{3}{4}}}{4} \]
T(45).✓ Proved
\[ \frac{20 \ln{\left(15 \right)}}{\ln{\left(2 \right)}} \]
Solve T(t) = target: e^(−kt) = (target − Tₐ)/(T₀ − Tₐ), so t = ln((T₀ − Tₐ)/(target − Tₐ))/k.✓ Proved

Answer \( T(45) = 22 + \frac{75 \cdot 2^{\frac{3}{4}}}{4} \approx 53.53^\circ,\quad t = \frac{20 \ln{\left(15 \right)}}{\ln{\left(2 \right)}} \approx 78.14\text{ min} \)

Lines: 4 proved, 1 reviewed. 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	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 provides the correct formulas and numerical approximations for both requested values.

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 provides the correct formulas and numerical approximations for both requested values.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution contains a critical arithmetic error in determining the constant k. The temperature difference drops from 150 to 75, which is a factor
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.