Exponential growth and decay
Problem 5.288 · medium
A bacteria culture starts with 100 cells and doubles every 3 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)/3.
- \[ 400 \cdot 2^{1 \cdot \frac{1}{3}} = 400 \sqrt[3]{2} \]y(7).✓ Proved
- \[ \frac{3 \ln{\left(5 \right)}}{\ln{\left(2 \right)}} \]Solve y₀e^(kt) = 500 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(7) = 400 \sqrt[3]{2} \approx 503.97,\quad t = \frac{3 \ln{\left(5 \right)}}{\ln{\left(2 \right)}} \approx 6.966 \)
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 fails to explicitly derive the formula y(t) = 100 * 2^(t/3) or y(t) = 100 * e^(kt) with the correct k, jumping directly to the numerical answer for y(7) without showing the substitution t=7 into the model. Additionally, step 2 defines k but does not explicitly state the resulting function y(t) used in step 3, making the logical connection between the parameter k and the final calculation opaque.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to explicitly derive the formula y(t) = 100 * 2^(t/3) or y(t) = 100 * e^(kt) with the correct k, jumping directly to the numerical answer for y(7) without showing the substitution t=7 into the model. Additionally, step 2 defines k but does not explicitly state the resulting function y(t) used in step 3, making the logical connection between the parameter k and the final calculation opaque.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to explicitly derive the formula for k using natural logarithms (ln) as required by the exponential model y = y₀e^{kt}, instead stating k = log(2)/3 without specifying the base, which is ambiguous and technically incorrect if 'log' implies base 10. Furthermore, the derivation of t is skipped entirely, jumping from the model to the final answer without showing the algebraic steps or verifying that the hypotheses for the exponential growth model are met.
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.