Exponential growth and decay
Problem 5.359 · medium
A bacteria culture starts with 1000 cells and doubles every 2 hours. Find the number of cells after 5 hours, and how long it takes to reach 3000 cells.
- y = y₀e^{kt}; the given doubling time, half-life or second data point fixes k.Reviewed
- k = log(2)/2.Reviewed
- \[ 4000 \sqrt{2} \]y(5).✓ Proved
- \[ \frac{2 \ln{\left(3 \right)}}{\ln{\left(2 \right)}} \]Solve y₀e^(kt) = 3000 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(5) = 4000 \sqrt{2} \approx 5656.9,\quad t = \frac{2 \ln{\left(3 \right)}}{\ln{\left(2 \right)}} \approx 3.17 \)
Lines: 2 proved, 2 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the exponential growth model, derives the rate constant k from the doubling time, and provides the correct symbolic and numerical answers for both parts of the problem.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the exponential growth model, derives the rate constant k from the doubling time, and provides the correct symbolic and numerical answers for both parts of the problem.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly define the initial condition y₀ = 1000, which is necessary to derive the specific function used in steps 3 and 4. Without stating y₀, the jump from the general formula to the specific numerical answers is logically incomplete.
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.