Exponential growth and decay
Problem 5.499 · medium
A quantity grows exponentially: it is 400 at \( \displaystyle t = 0 \) and 1000 at \( \displaystyle t = 2 \). Find it at \( \displaystyle t = 4 \), and when it reaches 4000.
- y = y₀e^{kt}; the given doubling time, half-life or second data point fixes k.
- k = log(5/2)/2.
- \[ 2500 \]y(4).✓ Proved
- \[ \frac{2 \ln{\left(10 \right)}}{\ln{\left(\frac{5}{2} \right)}} = \ln{\left(10^{\frac{2}{\ln{\left(\frac{5}{2} \right)}}} \right)} \]Solve y₀e^(kt) = 4000 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(4) = 2500 \approx 2500,\quad t = \ln{\left(10^{\frac{2}{\ln{\left(\frac{5}{2} \right)}}} \right)} \approx 5.026 \)
Lines: 2 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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: fail (misleading) — The solution’s final sentence is unclear and does not correctly state the time t in terms of the given parameters; it mixes an identity with the actual formula, which could mislead a student about how to solve for t.qwen3.6:27b-mlx: fail (error) — The solution incorrectly refers to the second data point as a 'doubling time' or 'half-life' in step 1, which is factually wrong (the quantity grows from 400 to 1000, not doubling). Additionally, step 4 presents an identity rather than the actual solution for t, failing to explicitly state the final answer for the time when y=4000.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-12 — The solution’s final sentence is unclear and does not correctly state the time t in terms of the given parameters; it mixes an identity with the actual formula, which could mislead a student about how to solve for t.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly refers to the second data point as a 'doubling time' or 'half-life' in step 1, which is factually wrong (the quantity grows from 400 to 1000, not doubling). Additionally, step 4 presents an identity rather than the actual solution for t, failing to explicitly state the final answer for the time when y=4000.gpt-oss:20b: inconclusive 2026-10-12 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The computed value y(4) is incorrect. Using y(4)=400e^{4k}=400e^{2\ln(5/2)}=400(5/2)=1000, not 2500. The subsequent t‑formula is correct, but the earlierqwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly labels the given data point as a 'doubling time' or 'half-life' in step 1, which is factually wrong (the quantity grows from 400 to 1000, a factor of 2.5, not 2 or 0.5). Additionally, step 4 presents an unnecessarily complex and non-standard form for the time t, obscuring the simple logarithmic solution.
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 2026-10-12 with SymPy 1.14.0.