The logistic equation
Problem 6.349 · hard
A population follows the logistic model with growth rate \( \displaystyle r = \frac{1}{5} \) per year, carrying capacity 5000, and initial population 500. Find \( \displaystyle P(t) \), \( \displaystyle P(2) \), and when the population reaches 2500.
- Equilibria: P = 0 and P = K = 5000. The solution has the form P = K/(1 + Ae^(−rt)) with A = (K − P₀)/P₀.
- \[ 9 \]A.✓ Proved
- \[ \frac{d}{d t} \frac{5000}{1 + 9 e^{- \frac{t}{5}}} = \frac{1000 - \frac{1000}{1 + 9 e^{- \frac{t}{5}}}}{1 + 9 e^{- \frac{t}{5}}} \]This P(t) satisfies the differential equation.✓ Proved
- \[ 500 \]and the initial condition.✓ Proved
- \[ \frac{5000}{1 + \frac{9}{e^{\frac{2}{5}}}} = \frac{5000 e^{\frac{2}{5}}}{e^{\frac{2}{5}} + 9} \]P(2).✓ 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 \( P(t) = \frac{5000}{1 + 9 e^{- \frac{t}{5}}},\ P(2) \approx 710.9,\ t = \ln{\left(59049 \right)} \approx 10.986 \)
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 numerical answer for P(2) as requested by the problem, providing only an algebraic simplification. Additionally, the calculation for t is incorrect: t = 5 * ln(9) ≈ 11.16, but the solution claims t = ln(59049) ≈ 10.986 (which is 5 * ln(9) is false, ln(59049) is 10.986, but 5*ln(9) is 11.16; wait, 9^5 = 59049, so ln(59049) = 5*ln(9). The value 10.986 is actually 5*ln(9)? Let's check. ln(9) ≈ 2.197. 5*2.197 = 10.985. Okay, the math is correct. The error is that the problem asks for P(2) and the solution provides an algebraic form but not the numerical approximation ~710.9 stated in the answer key, nor does it explicitly calculate it in the steps. However, looking closer at step 6, it says t = ln(59049). The problem asks for 'when the population reaches 2500'. The solution derives t = ln(A)/r = ln(9)/(1/5) = 5ln(9) = ln(9^5) = ln(59049). This is correct. The main issue is that Step 5 does not provide the numerical value for P(2) which is part of the question. The stated answer includes P(2) approx 710.9. The solution stops at the algebraic form. This is a completeness error.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to explicitly state the final numerical answer for P(2) as requested by the problem, providing only an algebraic simplification. Additionally, the calculation for t is incorrect: t = 5 * ln(9) ≈ 11.16, but the solution claims t = ln(59049) ≈ 10.986 (which is 5 * ln(9) is false, ln(59049) is 10.986, but 5*ln(9) is 11.16; wait, 9^5 = 59049, so ln(59049) = 5*ln(9). The value 10.986 is actually 5*ln(9)? Let's check. ln(9) ≈ 2.197. 5*2.197 = 10.985. Okay, the math is correct. The error is that the problem asks for P(2) and the solution provides an algebraic form but not the numerical approximation ~710.9 stated in the answer key, nor does it explicitly calculate it in the steps. However, looking closer at step 6, it says t = ln(59049). The problem asks for 'when the population reaches 2500'. The solution derives t = ln(A)/r = ln(9)/(1/5) = 5ln(9) = ln(9^5) = ln(59049). This is correct. The main issue is that Step 5 does not provide the numerical value for P(2) which is part of the question. The stated answer includes P(2) approx 710.9. The solution stops at the algebraic form. This is a completeness error.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the logistic model parameters, derives the constant A, and computes the specific values for P(2) and the time to reach half the carrying capacity. The algebraic steps are verified, and the final approximations are consistent with the exact forms.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.