Exponential growth and decay
Problem 5.360 · medium
A bacteria culture starts with 200 cells and doubles every 3 hours. Find the number of cells after 12 hours, and how long it takes to reach 2000 cells.
- y = y₀e^{kt}; the given doubling time, half-life or second data point fixes k.
- k = log(2)/3.
- \[ 3200 \]y(12).✓ Proved
- \[ \frac{3 \ln{\left(10 \right)}}{\ln{\left(2 \right)}} = \ln{\left(10^{\frac{3}{\ln{\left(2 \right)}}} \right)} \]Solve y₀e^(kt) = 2000 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(12) = 3200 \approx 3200,\quad t = \ln{\left(10^{\frac{3}{\ln{\left(2 \right)}}} \right)} \approx 9.966 \)
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) — The solution fails to explicitly state the initial condition y_0 = 200, which is necessary to derive the specific model y = 200e^{kt}. Without this, the calculation of y(12) and t is not justified by the preceding sentences.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly state the initial condition y_0 = 200, which is necessary to derive the specific model y = 200e^{kt}. Without this, the calculation of y(12) and t is not justified by the preceding sentences.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07
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.