Newton's law of cooling
Problem 6.319 · hard
An object at 75° is placed in a room at 0°. After 15 minutes it has cooled to \frac{75}{2}°. 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
- \[ \frac{d}{d t} 75 e^{- \frac{t \ln{\left(2 \right)}}{15}} = - 5 e^{- \frac{t \ln{\left(2 \right)}}{15}} \ln{\left(2 \right)} \]The model satisfies the cooling law.✓ Proved
- \[ \frac{75}{2} \]k = ln((T₀ − Tₐ)/(T₁ − Tₐ))/t₁ = log(2)/15 matches the reading at t = 15.✓ Proved
- \[ \frac{75}{4} \]T(30).✓ Proved
- \[ \frac{15 \ln{\left(15 \right)}}{\ln{\left(2 \right)}} \]Solve T(t) = target: e^(−kt) = (target − Tₐ)/(T₀ − Tₐ), so t = ln((T₀ − Tₐ)/(target − Tₐ))/k.✓ Proved
Answer \( T(30) = \frac{75}{4} \approx 18.75^\circ,\quad t = \frac{15 \ln{\left(15 \right)}}{\ln{\left(2 \right)}} \approx 58.60\text{ min} \)
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: passqwen3.6:27b-mlx: pass — The solution correctly applies Newton's Law of Cooling, derives the cooling constant k, and accurately computes the temperature at t=30 and the time to reach 5 degrees.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies Newton's Law of Cooling, derives the cooling constant k, and accurately computes the temperature at t=30 and the time to reach 5 degrees.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to explicitly state the derived value of k or the specific function T(t) before evaluating it, making the algebraic steps in lines 2-5 appear as ungrounded identities rather than a logical derivation. Furthermore, the final answer for the time to reach 5° is incorrect; the calculation yields t = 15 * ln(15) / ln(2) ≈ 58.6, but the correct time to reach 5° is t = 15 * ln(15) / ln(2) is wrong because T(t) = 5 implies 5 = 75 * 2^(-t/15) => 1/15 = 2^(-t/15) => -ln(15) = (-t/15)ln(2) => t = 15 ln(15)/ln(2). Wait, let me re-check the math. T(t) = 75 * 2^(-t/15). Set T(t)=5. 5 = 75 * 2^(-t/15). 1/15 = 2^(-t/15). ln(1/15) = (-t/15)ln(2). -ln(15) = (-t/15)ln(2). t = 15 ln(15)/ln(2). The stated answer is actually correct numerically. However, the solution text is extremely sparse and lacks the necessary explanatory sentences to justify the jump from the general law to the specific constants and final answers. It presents equations as 'unchecked' identities without showing the substitution. But more importantly, line 1 says 'solves dT/dt = -k(T-Ta)'. This is correct. Line 2 checks the derivative. Line 3 checks k. Line 4 checks T(30). Line 5 checks the time. The 'error' is that the solution is just a list of checks without the actual derivation steps being presented as sentences or logical flow. It's a 'style' issue at best, but the prompt asks to find what is wrong. Is there a mathematical error? Let's re-read carefully. T(30) = 75/4 = 18.75. Correct. t for 5 degrees: 15 ln(15)/ln(2). Correct. The solution is mathematically correct but presented as a series of tautologies/identities rather than a solution. However, usually 'fail' is reserved for mathematical errors. Let's look closer. Is there a subtle error? 'An object at 75... room at 0'. T_a = 0. T_0 = 75. T(15) = 37.5. k = ln(2)/15. T(t) = 75 e^(-kt). T(30) = 75 e^(-2 ln 2) = 75/4. Correct. T(t)=5 => 5 = 75 e^(-kt) => 1/15 = e^(-kt) => -ln 15 = -kt => t = ln 15 / k = 15 ln 15 / ln 2. Correct. The solution is correct. Why would it fail? Maybe the prompt implies I should find a flaw. Let's look at the 'unchecked' tags. The user says 'Each line... is either an EQUATION... or a SENTENCE'. The solution provides sentences only in line 1. Lines 2-5 are equations. The review instructions say 'Check that... each conclusion follows from the lines before it'. Line 2 follows from Line 1? No, Line 2 is a specific instance. Line 1 is general. The specific instance isn't derived in text. This is a 'style' or 'misleading' issue because it doesn't teach the student *how* to get there, just that the answer is consistent. But is it 'wrong'? No. Is it 'misleading'? Yes, it suggests that checking the answer is the same as solving it. I will mark it as 'misleading' because it fails to model the problem-solving process, only the verification process.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.