Exponential growth and decay
Problem 5.501 · medium
A bacteria culture starts with 500 cells and doubles every 4 hours. Find the number of cells after 1 hours, and how long it takes to reach 2500 cells.
- y = y₀e^{kt}; the given doubling time, half-life or second data point fixes k.
- k = log(2)/4.
- \[ 500 \cdot 2^{1 \cdot \frac{1}{4}} = 500 \sqrt[4]{2} \]y(1).✓ Proved
- \[ \frac{4 \ln{\left(5 \right)}}{\ln{\left(2 \right)}} \]Solve y₀e^(kt) = 2500 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(1) = 500 \sqrt[4]{2} \approx 594.6,\quad t = \frac{4 \ln{\left(5 \right)}}{\ln{\left(2 \right)}} \approx 9.288 \)
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 (error) — The solution uses 'log' in step 2 without specifying the base, which is ambiguous and potentially incorrect if interpreted as base 10. Step 4 explicitly uses 'ln' (natural log), creating an inconsistency. The formula for k should explicitly use natural logarithms (ln(2)/4) to match the exponential model y = y₀e^{kt}.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution uses 'log' in step 2 without specifying the base, which is ambiguous and potentially incorrect if interpreted as base 10. Step 4 explicitly uses 'ln' (natural log), creating an inconsistency. The formula for k should explicitly use natural logarithms (ln(2)/4) to match the exponential model y = y₀e^{kt}.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (misleading) 2026-10-12 — The solution uses 'log' in step 2 without specifying the base, which is ambiguous in calculus contexts where 'ln' is standard. While the final answer uses 'ln', the intermediate step is imprecise and could confuse a student about the relationship between doubling time and the rate constant k.
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 2026-10-12 with SymPy 1.14.0.