The logistic equation
Problem 6.384 · hard
A population follows the logistic model with growth rate \( \displaystyle r = \frac{1}{10} \) per year, carrying capacity 500, and initial population 50. 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.
- \[ 9 \]A.✓ Proved
- \[ \frac{d}{d t} \frac{500}{1 + 9 e^{- \frac{t}{10}}} = \frac{50 - \frac{50}{1 + 9 e^{- \frac{t}{10}}}}{1 + 9 e^{- \frac{t}{10}}} \]This P(t) satisfies the differential equation.✓ Proved
- \[ 50 \]and the initial condition.✓ Proved
- \[ \frac{500}{1 + \frac{9}{e^{\frac{3}{5}}}} = \frac{500 e^{\frac{3}{5}}}{e^{\frac{3}{5}} + 9} \]P(6).✓ Proved
- \[ 10 \ln{\left(9 \right)} = \ln{\left(3486784401 \right)} \]Half the carrying capacity when Ae^(−rt) = 1, i.e. t = ln(A)/r.✓ Proved
Answer \( P(t) = \frac{500}{1 + 9 e^{- \frac{t}{10}}},\ P(6) \approx 84.2,\ t = \ln{\left(3486784401 \right)} \approx 21.972 \)
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(6) requested in the problem statement, only providing the exact symbolic form. Additionally, the final answer for t is given as ln(3486784401), which is incorrect; the correct value is 10*ln(9) ≈ 21.97, whereas ln(3486784401) ≈ 21.97 is a coincidence of magnitude but the expression ln(9^10) is not simplified to 10ln(9) in the final answer format, and more importantly, the problem asks for 'when', implying a time value, but the provided answer format is ambiguous and the intermediate step 6 claims 10*log(9) = log(3486784401) which is true, but the final answer lists t = ln(3486784401) which is technically correct but unusual, however the main error is the missing numerical value for P(6) in the final summary or the solution flow.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to provide the numerical approximation for P(6) requested in the problem statement, only providing the exact symbolic form. Additionally, the final answer for t is given as ln(3486784401), which is incorrect; the correct value is 10*ln(9) ≈ 21.97, whereas ln(3486784401) ≈ 21.97 is a coincidence of magnitude but the expression ln(9^10) is not simplified to 10ln(9) in the final answer format, and more importantly, the problem asks for 'when', implying a time value, but the provided answer format is ambiguous and the intermediate step 6 claims 10*log(9) = log(3486784401) which is true, but the final answer lists t = ln(3486784401) which is technically correct but unusual, however the main error is the missing numerical value for P(6) in the final summary or the solution flow.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution claims t = ln(3486784401) ≈ 21.972, but ln(3486784401) is approximately 21.97, while the correct time to reach half capacity is t = 10*ln(9) ≈ 21.972. The value 3486784401 is 9^10, so ln(9^10) = 10*ln(9), which is correct. However, the notation 'ln(3486784401)' is misleadingly precise and implies an exact integer argument for a transcendental function, whereas the standard form is 10 ln 9. More critically, the prompt asks for P(6). The solution calculates P(6) in step 5 but does not provide the numerical approximation 84.2 in the text, only the symbolic form. The stated answer includes P(6) ≈ 84.2, but the solution steps do not derive or state this number, making the connection between the steps and the final answer opaque. Furthermore, step 6 says '10*log(9) = log(3486784401)', which is true, but the final answer lists t = ln(3486784401). This is technically correct but stylistically poor. The main issue is that the solution does not explicitly calculate P(6) ≈ 84.2, leaving a gap between the symbolic expression and the stated numerical answer.gpt-oss:20b: pass 2026-10-12
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.