The logistic equation
Problem 6.342 · hard
A population follows the logistic model with growth rate \( \displaystyle r = \frac{1}{5} \) per year, carrying capacity 500, and initial population 25. Find \( \displaystyle P(t) \), \( \displaystyle P(6) \), and when the population reaches 250.
- Equilibria: P = 0 and P = K = 500. The solution has the form P = K/(1 + Ae^(−rt)) with A = (K − P₀)/P₀.
- \[ 19 \]A.✓ Proved
- \[ \frac{d}{d t} \frac{500}{1 + 19 e^{- \frac{t}{5}}} = \frac{100 - \frac{100}{1 + 19 e^{- \frac{t}{5}}}}{1 + 19 e^{- \frac{t}{5}}} \]This P(t) satisfies the differential equation.✓ Proved
- \[ 25 \]and the initial condition.✓ Proved
- \[ \frac{500}{1 + \frac{19}{e^{\frac{6}{5}}}} = \frac{500 e^{\frac{6}{5}}}{e^{\frac{6}{5}} + 19} \]P(6).✓ Proved
- \[ 5 \ln{\left(19 \right)} = \ln{\left(2476099 \right)} \]Half the carrying capacity when Ae^(−rt) = 1, i.e. t = ln(A)/r.✓ Proved
Answer \( P(t) = \frac{500}{1 + 19 e^{- \frac{t}{5}}},\ P(6) \approx 74.4,\ t = \ln{\left(2476099 \right)} \approx 14.722 \)
Lines: 5 proved, 1 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | ✓ 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 differential equation solved numerically (Taylor-series ODE solver) from P(0) |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the final answer for P(6) as a numerical value, despite the problem asking for it and the stated answer providing an approximation. Additionally, the final line presents an exact symbolic equality for t (5*log(19) = log(2476099)) but does not explicitly state the final numerical answer for t (~14.722) as required by the problem statement.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to explicitly state the final answer for P(6) as a numerical value, despite the problem asking for it and the stated answer providing an approximation. Additionally, the final line presents an exact symbolic equality for t (5*log(19) = log(2476099)) but does not explicitly state the final numerical answer for t (~14.722) as required by the problem statement.qwen3.6:27b-mlx: inconclusive 2026-10-10 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution claims t = ln(2476099) ≈ 14.722, but ln(2476099) is approximately 14.722 only if the base is e, yet the value 2476099 is 19^5, so ln(19^gpt-oss:20b: pass 2026-10-10
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/logistic_equation, checked 2026-10-10 with SymPy 1.14.0.