The logistic equation
Problem 6.269 · hard
A population follows the logistic model with growth rate \( \displaystyle r = \frac{1}{10} \) per year, carrying capacity 100, and initial population 2. Find \( \displaystyle P(t) \), \( \displaystyle P(8) \), and when the population reaches 50.
- Equilibria: P = 0 and P = K = 100. The solution has the form P = K/(1 + Ae^(−rt)) with A = (K − P₀)/P₀.
- \[ 49 \]A.✓ Proved
- \[ \frac{d}{d t} \frac{100}{1 + 49 e^{- \frac{t}{10}}} = \frac{10 - \frac{10}{1 + 49 e^{- \frac{t}{10}}}}{1 + 49 e^{- \frac{t}{10}}} \]This P(t) satisfies the differential equation.✓ Proved
- \[ 2 \]and the initial condition.✓ Proved
- \[ \frac{100}{1 + \frac{49}{e^{\frac{4}{5}}}} = \frac{100 e^{\frac{4}{5}}}{e^{\frac{4}{5}} + 49} \]P(8).✓ Proved
- \[ 10 \ln{\left(49 \right)} = \ln{\left(79792266297612001 \right)} \]Half the carrying capacity when Ae^(−rt) = 1, i.e. t = ln(A)/r.✓ Proved
Answer \( P(t) = \frac{100}{1 + 49 e^{- \frac{t}{10}}},\ P(8) \approx 4.3,\ t = \ln{\left(79792266297612001 \right)} \approx 38.918 \)
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 provide the numerical approximation for P(8) requested in the problem statement, leaving only the exact form. Additionally, the final line presents an exact equality between a transcendental number and an integer, which is mathematically false, rather than stating the approximation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to provide the numerical approximation for P(8) requested in the problem statement, leaving only the exact form. Additionally, the final line presents an exact equality between a transcendental number and an integer, which is mathematically false, rather than stating the approximation.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution fails to provide the numerical approximation for P(8) requested in the problem statement, only showing the symbolic expression. Additionally, the final answer for t is presented as the exact logarithm rather than the required approximate value, and the logic in step 6 is disjointed.gpt-oss:20b: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.