Exponential growth and decay
Problem 5.424 · medium
A bacteria culture starts with 100 cells and doubles every 3 hours. Find the number of cells after 2 hours, and how long it takes to reach 300 cells.
- y = y₀e^{kt}; the given doubling time, half-life or second data point fixes k.Reviewed
- k = log(2)/3.Reviewed
- \[ 100 \cdot 2^{\frac{2}{3}} \]y(2).✓ Proved
- \[ \frac{3 \ln{\left(3 \right)}}{\ln{\left(2 \right)}} \]Solve y₀e^(kt) = 300 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(2) = 100 \cdot 2^{\frac{2}{3}} \approx 158.74,\quad t = \frac{3 \ln{\left(3 \right)}}{\ln{\left(2 \right)}} \approx 4.755 \)
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 sets up the exponential growth model, derives the rate constant k, and provides the correct formulas and numerical approximations for both parts of the problem.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly sets up the exponential growth model, derives the rate constant k, and provides the correct formulas and numerical approximations for both parts of the problem.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the exponential growth model, calculates the rate constant k using the doubling time, and applies the formula to find both the population at t=2 and the time to reach 300 cells. The algebraic steps are correct and match the stated answer.
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.