Newton's law of cooling
Problem 6.396 · hard
An object at 50° is placed in a room at 0°. After 5 minutes it has cooled to \frac{75}{2}°. Using Newton's law of cooling, find its temperature after 15 minutes and when it reaches 15°.
- T(t) = Tₐ + (T₀ − Tₐ)e^(−kt) solves dT/dt = −k(T − Tₐ).
- \[ \frac{d}{d t} 50 e^{- \frac{t \ln{\left(\frac{4}{3} \right)}}{5}} = - 10 e^{- \frac{t \ln{\left(\frac{4}{3} \right)}}{5}} \ln{\left(\frac{4}{3} \right)} \]The model satisfies the cooling law.✓ Proved
- \[ \frac{75}{2} \]k = ln((T₀ − Tₐ)/(T₁ − Tₐ))/t₁ = log(4/3)/5 matches the reading at t = 5.✓ Proved
- \[ \frac{675}{32} \]T(15).✓ Proved
- \[ \frac{5 \ln{\left(\frac{10}{3} \right)}}{\ln{\left(\frac{4}{3} \right)}} = \ln{\left(\left(\frac{10}{3}\right)^{\frac{5}{\ln{\left(\frac{4}{3} \right)}}} \right)} \]Solve T(t) = target: e^(−kt) = (target − Tₐ)/(T₀ − Tₐ), so t = ln((T₀ − Tₐ)/(target − Tₐ))/k.✓ Proved
Answer \( T(15) = \frac{675}{32} \approx 21.09^\circ,\quad t = \ln{\left(\left(\frac{10}{3}\right)^{\frac{5}{\ln{\left(\frac{4}{3} \right)}}} \right)} \approx 20.93\text{ min} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly define the function T(t) with the calculated constant k, making the subsequent verification steps (lines 2-5) logically disconnected from the problem statement. Line 2 verifies a specific function but does not state that this function is the solution derived from the given conditions, leaving the link between the general law and the specific instance implicit and formally incomplete.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to explicitly define the function T(t) with the calculated constant k, making the subsequent verification steps (lines 2-5) logically disconnected from the problem statement. Line 2 verifies a specific function but does not state that this function is the solution derived from the given conditions, leaving the link between the general law and the specific instance implicit and formally incomplete.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to explicitly define the function T(t) with the calculated constant k, making the subsequent verification steps (lines 2-5) logically disconnected from the problem statement. Furthermore, the final answer for time is presented in a mathematically redundant form (ln(x^y) instead of y*ln(x)) and the derivation steps are presented as tautological identity checks rather than a clear logical flow.
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.