The logistic equation
Problem 6.383 · hard
A population follows the logistic model with growth rate \( \displaystyle r = \frac{1}{5} \) per year, carrying capacity 10000, and initial population 1000. Find \( \displaystyle P(t) \), \( \displaystyle P(1) \), and when the population reaches 5000.
- Equilibria: P = 0 and P = K = 10000. The solution has the form P = K/(1 + Ae^(−rt)) with A = (K − P₀)/P₀.
- \[ 9 \]A.✓ Proved
- \[ \frac{d}{d t} \frac{10000}{1 + 9 e^{- \frac{t}{5}}} = \frac{2000 - \frac{2000}{1 + 9 e^{- \frac{t}{5}}}}{1 + 9 e^{- \frac{t}{5}}} \]This P(t) satisfies the differential equation.✓ Proved
- \[ 1000 \]and the initial condition.✓ Proved
- \[ \frac{10000}{1 + \frac{9}{e^{\frac{1}{5}}}} = \frac{10000 e^{\frac{1}{5}}}{e^{\frac{1}{5}} + 9} \]P(1).✓ Proved
- \[ 5 \ln{\left(9 \right)} = \ln{\left(59049 \right)} \]Half the carrying capacity when Ae^(−rt) = 1, i.e. t = ln(A)/r.✓ Proved
Answer \( P(t) = \frac{10000}{1 + 9 e^{- \frac{t}{5}}},\ P(1) \approx 1194.9,\ t = \ln{\left(59049 \right)} \approx 10.986 \)
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(1) as a numerical value, despite the problem asking for it and the stated answer providing an approximation. Additionally, the final line calculates t = ln(59049) but omits the division by r (or multiplication by 5) required to get the correct time, although the stated answer implies the correct value was intended, the derivation shown is incomplete/incorrect for finding t.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to explicitly state the final answer for P(1) as a numerical value, despite the problem asking for it and the stated answer providing an approximation. Additionally, the final line calculates t = ln(59049) but omits the division by r (or multiplication by 5) required to get the correct time, although the stated answer implies the correct value was intended, the derivation shown is incomplete/incorrect for finding t.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly identifies the logistic growth formula, calculates the constant A, verifies the differential equation and initial condition, and correctly solves for 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.