Exponential growth and decay
Problem 5.361 · medium
A quantity grows exponentially: it is 400 at \( \displaystyle t = 0 \) and 1200 at \( \displaystyle t = 5 \). Find it at \( \displaystyle t = 10 \), and when it reaches 4000.
- y = y₀e^{kt}; the given doubling time, half-life or second data point fixes k.
- k = log(3)/5.
- \[ 3600 \]y(10).✓ Proved
- \[ \frac{5 \ln{\left(10 \right)}}{\ln{\left(3 \right)}} \]Solve y₀e^(kt) = 4000 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(10) = 3600 \approx 3600,\quad t = \frac{5 \ln{\left(10 \right)}}{\ln{\left(3 \right)}} \approx 10.48 \)
Lines: 2 proved, 2 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 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ 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 | the model evaluated, and the time found by a numerical root-finder |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — Line 1 incorrectly refers to a 'doubling time' when the quantity triples (400 to 1200). Line 2 uses 'log' without specifying the base, which is ambiguous and potentially incorrect depending on interpretation, though the numerical result in line 4 implies natural log was used.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — Line 1 incorrectly refers to a 'doubling time' when the quantity triples (400 to 1200). Line 2 uses 'log' without specifying the base, which is ambiguous and potentially incorrect depending on interpretation, though the numerical result in line 4 implies natural log was used.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution incorrectly labels the second data point as a 'doubling time' or 'half-life', which are specific cases not applicable here (the quantity triples). Additionally, the final time calculation for reaching 4000 is incorrect; it should be t = 5 * ln(10) / ln(3), but the solution presents the formula t = ln(M/y0)/k without substituting k correctly in the text description, although the final numerical value is derived from the correct expression. The primary error is the conceptual mislabeling of the growth factor.
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/exponential_models, checked 2026-10-07 with SymPy 1.14.0.