The logistic equation
Problem 6.385 · hard
A population follows the logistic model with growth rate \( \displaystyle r = \frac{1}{10} \) per year, carrying capacity 500, and initial population 100. Find \( \displaystyle P(t) \), \( \displaystyle P(1) \), 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.
- \[ 4 \]A.✓ Proved
- \[ \frac{d}{d t} \frac{500}{1 + 4 e^{- \frac{t}{10}}} = \frac{50 - \frac{50}{1 + 4 e^{- \frac{t}{10}}}}{1 + 4 e^{- \frac{t}{10}}} \]This P(t) satisfies the differential equation.✓ Proved
- \[ 100 \]and the initial condition.✓ Proved
- \[ \frac{500}{1 + \frac{4}{e^{\frac{1}{10}}}} = \frac{500 e^{\frac{1}{10}}}{e^{\frac{1}{10}} + 4} \]P(1).✓ Proved
- \[ 10 \ln{\left(4 \right)} = \ln{\left(1048576 \right)} \]Half the carrying capacity when Ae^(−rt) = 1, i.e. t = ln(A)/r.✓ Proved
Answer \( P(t) = \frac{500}{1 + 4 e^{- \frac{t}{10}}},\ P(1) \approx 108.2,\ t = \ln{\left(1048576 \right)} \approx 13.863 \)
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: fail (misleading) — The solution omits the numerical value of P(1) and does not explicitly state that t≈13.863 is when the population reaches 250. The steps are correct, but the final conclusions are missing, which could mislead a student into thinking the answer is complete.qwen3.6:27b-mlx: fail (error) — The solution claims t = ln(1048576) ≈ 13.863, but ln(1048576) ≈ 13.857. The value 13.863 corresponds to 10*ln(4), not ln(1048576). The equation in step 6 is also mathematically false (10*ln(4) ≠ ln(1048576)).
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-12 — The solution omits the numerical value of P(1) and does not explicitly state that t≈13.863 is when the population reaches 250. The steps are correct, but the final conclusions are missing, which could mislead a student into thinking the answer is complete.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution claims t = ln(1048576) ≈ 13.863, but ln(1048576) ≈ 13.857. The value 13.863 corresponds to 10*ln(4), not ln(1048576). The equation in step 6 is also mathematically false (10*ln(4) ≠ ln(1048576)).gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly derives the logistic function, verifies the differential equation and initial condition, and accurately computes P(1) and the time to reach half the carrying capacity.
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 2026-10-12 with SymPy 1.14.0.