Exponential growth and decay
Problem 5.287 · medium
A bacteria culture starts with 100 cells and doubles every 5 hours. Find the number of cells after 10 hours, and how long it takes to reach 1000 cells.
- y = y₀e^{kt}; the given doubling time, half-life or second data point fixes k.
- k = log(2)/5.
- \[ 400 \]y(10).✓ Proved
- \[ \frac{5 \ln{\left(10 \right)}}{\ln{\left(2 \right)}} = \ln{\left(10^{\frac{5}{\ln{\left(2 \right)}}} \right)} \]Solve y₀e^(kt) = 1000 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(10) = 400 \approx 400,\quad t = \ln{\left(10^{\frac{5}{\ln{\left(2 \right)}}} \right)} \approx 16.61 \)
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 (style) — The solution is mathematically correct but stylistically poor; it presents the final numerical answers as proven equations without showing the derivation steps, making it unclear how the results were obtained.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (style) 2026-10-06 — The solution is mathematically correct but stylistically poor; it presents the final numerical answers as proven equations without showing the derivation steps, making it unclear how the results were obtained.gpt-oss:20b: inconclusive 2026-10-06 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution does not correctly compute the number of cells at 10 h (it merely states 400=400 without showing the calculation) and the time‑to‑1000 cellsqwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly models exponential growth, determines the rate constant k, and computes the correct values for y(10) and the time t to reach 1000 cells. The algebraic identity in line 4 is valid, though the notation is slightly unconventional.
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 2026-10-06 with SymPy 1.14.0.