Exponential growth and decay
Problem 5.284 · medium
A bacteria culture starts with 50 cells and doubles every 6 hours. Find the number of cells after 7 hours, and how long it takes to reach 500 cells.
- y = y₀e^{kt}; the given doubling time, half-life or second data point fixes k.
- k = log(2)/6.
- \[ 100 \cdot 2^{1 \cdot \frac{1}{6}} = 100 \sqrt[6]{2} \]y(7).✓ Proved
- \[ \frac{6 \ln{\left(10 \right)}}{\ln{\left(2 \right)}} = \ln{\left(10^{\frac{6}{\ln{\left(2 \right)}}} \right)} \]Solve y₀e^(kt) = 500 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(7) = 100 \sqrt[6]{2} \approx 112.25,\quad t = \ln{\left(10^{\frac{6}{\ln{\left(2 \right)}}} \right)} \approx 19.93 \)
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: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution uses log base 10 for k in line 2, but line 4 uses natural log for the time calculation, creating an inconsistency. Additionally, the fin
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution uses log base 10 for k in line 2, but line 4 uses natural log for the time calculation, creating an inconsistency. Additionally, the fingpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution incorrectly calculates the time to reach 500 cells. The correct time is approximately 18.03 hours, but the solution yields ~19.93 hours due to an algebraic error in the final expression (likely confusing base 10 and natural log or misapplying the exponent). Additionally, the intermediate step for y(7) is presented as an unchecked identity rather than a derived value, and the final answer format is confusing.
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.