Exponential growth and decay
Problem 5.290 · medium
A bacteria culture starts with 200 cells and doubles every 6 hours. Find the number of cells after 6 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)/6.
- \[ 400 \]y(6).✓ Proved
- \[ \frac{6 \ln{\left(5 \right)}}{\ln{\left(2 \right)}} \]Solve y₀e^(kt) = 1000 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(6) = 400 \approx 400,\quad t = \frac{6 \ln{\left(5 \right)}}{\ln{\left(2 \right)}} \approx 13.93 \)
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: fail (error) — The solution incorrectly states y(6)=400 without showing the calculation from y=200e^{k·6}. The equation 400=400 is trivial and does not verify the model. Additionally, the use of log(2) without specifying the base is ambiguous. The time calculation is correct in form but the steps are not properly justified.qwen3.6:27b-mlx: pass — The solution correctly identifies the exponential growth model, calculates the rate constant k, and derives the correct expressions for the population at t=6 and the time to reach 1000 cells.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — The solution incorrectly states y(6)=400 without showing the calculation from y=200e^{k·6}. The equation 400=400 is trivial and does not verify the model. Additionally, the use of log(2) without specifying the base is ambiguous. The time calculation is correct in form but the steps are not properly justified.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the exponential growth model, calculates the rate constant k, and derives the correct expressions for the population at t=6 and the time to reach 1000 cells.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to explicitly state the initial condition y₀ = 200, which is required to determine the constant in the exponential model. Without stating y₀, the derivation of k and the final time t lacks necessary context, making the logic 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 2026-10-06 with SymPy 1.14.0.