Exponential growth and decay
Problem 5.428 · medium
A quantity grows exponentially: it is 400 at \( \displaystyle t = 0 \) and 800 at \( \displaystyle t = 1 \). Find it at \( \displaystyle t = 2 \), and when it reaches 4000.
- y = y₀e^{kt}; the given doubling time, half-life or second data point fixes k.
- k = log(2).
- \[ 1600 \]y(2).✓ Proved
- \[ \frac{\ln{\left(10 \right)}}{\ln{\left(2 \right)}} = 1 + \frac{\ln{\left(5 \right)}}{\ln{\left(2 \right)}} \]Solve y₀e^(kt) = 4000 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(2) = 1600 \approx 1600,\quad t = 1 + \frac{\ln{\left(5 \right)}}{\ln{\left(2 \right)}} \approx 3.322 \)
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 (misleading) — The solution fails to explicitly state the base of the logarithm in step 2, which is critical for determining the value of k. While the final numerical answer implies natural logs were used, stating 'k = log(2)' without specifying 'ln' is ambiguous and could lead a student to use base-10 logs, resulting in an incorrect growth rate. Additionally, the setup in step 1 is vague about how the specific data points map to the parameters.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (misleading) 2026-10-09 — The solution fails to explicitly state the base of the logarithm in step 2, which is critical for determining the value of k. While the final numerical answer implies natural logs were used, stating 'k = log(2)' without specifying 'ln' is ambiguous and could lead a student to use base-10 logs, resulting in an incorrect growth rate. Additionally, the setup in step 1 is vague about how the specific data points map to the parameters.gpt-oss:20b: fail (error) 2026-10-09 — The second sentence claims k = log(2) without specifying the base; the correct value is k = ln 2, derived from e^k = 2 because the quantity doubles in one time unit. This omission misstates the parameter and could mislead a student about the role of logarithm bases.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to explicitly calculate the growth constant k using the given data points (y(0)=400, y(1)=800), instead stating k=log(2) without justification. While the numerical result for k is correct for base e, the logical step connecting the data to the parameter is missing, and the final answer format is ambiguous regarding the base of the logarithm.
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 2026-10-09 with SymPy 1.14.0.