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Newton's law of cooling

Problem 6.393 · hard

An object at 145° is placed in a room at -5°. After 10 minutes it has cooled to 70°. Using Newton's law of cooling, find its temperature after 30 minutes and when it reaches 5°.

T(t) = Tₐ + (T₀ − Tₐ)e^(−kt) solves dT/dt = −k(T − Tₐ).
Reviewed
\[ \frac{d}{d t} \left(-5 + 150 e^{- \frac{t \ln{\left(2 \right)}}{10}}\right) = - 15 e^{- \frac{t \ln{\left(2 \right)}}{10}} \ln{\left(2 \right)} \]
The model satisfies the cooling law.✓ Proved
\[ 70 \]
k = ln((T₀ − Tₐ)/(T₁ − Tₐ))/t₁ = log(2)/10 matches the reading at t = 10.✓ Proved
\[ \frac{55}{4} \]
T(30).✓ Proved
\[ \frac{10 \ln{\left(15 \right)}}{\ln{\left(2 \right)}} \]
Solve T(t) = target: e^(−kt) = (target − Tₐ)/(T₀ − Tₐ), so t = ln((T₀ − Tₐ)/(target − Tₐ))/k.✓ Proved

Answer \( T(30) = \frac{55}{4} \approx 13.75^\circ,\quad t = \frac{10 \ln{\left(15 \right)}}{\ln{\left(2 \right)}} \approx 39.07\text{ min} \)

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 computes the requested values for T(30) and the time to reach 5°. The algebraic steps are verified and the logic is sound.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: pass 2026-10-12 — 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(30) and the time to reach 5°. The algebraic steps are verified and the logic is sound.
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to explicitly state the derived temperature function T(t) = -5 + 150e^(-kt) with the calculated k, which is necessary to justify the subsequent evaluation steps. Additionally, the final answer for the time to reach 5° is incorrect; the object reaches 5° when T(t)=5, implying 5 = -5 + 150e^(-kt), so e^(-kt) = 10/150 = 1/15, leading to t = (10 ln 15)/ln 2, but the problem asks for when it reaches 5°, and the provided answer matches this calculation. Wait, let me re-read carefully. The problem asks for T(30) and when it reaches 5°. The solution provides T(30) = 13.75 and t = 39.07. Let's check the math. T_a = -5, T_0 = 145. T(10) = 70. 70 = -5 + 150 e^(-10k) => 75 = 150 e^(-10k) => 1/2 = e^(-10k) => -10k = ln(1/2) => k = ln(2)/10. Correct. T(30) = -5 + 150 e^(-3 ln(2)) = -5 + 150 (1/8) = -5 + 18.75 = 13.75. Correct. When T(t) = 5? 5 = -5 + 150 e^(-kt) => 10 = 150 e^(-kt) => 1/15 = e^(-kt) => -kt = ln(1/15) => t = ln(15)/k = 10 ln(15)/ln(2). Correct. The math is correct. Why did I think it was wrong? Ah, the prompt says 'find its temperature after 30 minutes and when it reaches 5°'. The solution calculates exactly that. Is there a subtle error? 'An object at 145°... room at -5°'. T_a = -5. T_0 = 145. T(10) = 70. The setup is correct. The derivation of k is correct. The final values are correct. The sentences are sparse but the equations are marked 'unchecked' by the user prompt description, but the instructions say 'Each line... is either an EQUATION... or a SENTENCE... The sentences are what you are here for.' Line 1 is a sentence. It states the general solution. It is correct. Lines 2-5 are equations. The prompt says 'do not re-check the algebra of a line marked proved'. Here they are marked 'unchecked' in the text provided by the user, but the instructions say 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. The user's input format is slightly ambiguous. It says '2. [equation, unchecked] ...'. This likely means the CAS hasn't checked it, or the user is presenting it as a fact. However, the core task is to review the *sentences*. Line 1 is the only sentence. It is correct. Are there missing sentences? The solution jumps from the general formula to specific evaluations without stating the specific model T(t) = -5 + 150e^(-t ln 2 / 10). This is a 'style' or 'clarity' issue, but is it an 'error'? The prompt asks to find what is wrong. If the math is correct, is it misleading? No. Is it an error? No. Wait, look at line 5. 'Solve T(t) = target... t = ...'. This is a sentence embedded in the equation line? No, it's labeled [equation]. The text after the equation is likely a comment. The instructions say 'Reply with JSON...'. If the solution is mathematically sound and the single sentence is correct, it should pass. However, often these adversarial reviews look for subtle sign errors or boundary issues. T_a = -5. T_0 = 145. Delta T_0 = 150. T(10) = 70. Delta T_10 = 75. Ratio 1/2. k = ln 2 / 10. T(30) -> Delta T_30 = 150 * (1/2)^3 = 150/8 = 18.75. T = -5 + 18.75 = 13.75. Correct. Time to 5: Delta T = 10. 10 = 150 * (1/2)^(t/10). 1/15 = (1/2)^(t/10). ln(1/15) = (t/10) ln(1/2). -ln 15 = (t/10) (-ln 2). t = 10 ln 15 / ln 2. Correct. The solution is correct. Why would it fail? Maybe the sentence 'T(t) = T_a + (T_0 - T_a)e^(-kt) solves dT/dt = -k(T - T_a)' is considered incomplete because it doesn't specify k > 0? Newton's law usually implies k > 0 for cooling. If T > T_a, dT/dt < 0, so -k(T-T_a) < 0, so k > 0. This is standard. I will mark it as pass.
gpt-oss:20b: pass 2026-10-12

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 2026-10-12 with SymPy 1.14.0.