Exponential growth and decay
Problem 5.432 · medium
A quantity grows exponentially: it is 200 at \( \displaystyle t = 0 \) and 500 at \( \displaystyle t = 4 \). Find it at \( \displaystyle t = 8 \), and when it reaches 2000.
- y = y₀e^{kt}; the given doubling time, half-life or second data point fixes k.
- k = log(5/2)/4.
- \[ 1250 \]y(8).✓ Proved
- \[ \frac{4 \ln{\left(10 \right)}}{\ln{\left(\frac{5}{2} \right)}} = \ln{\left(10^{\frac{4}{\ln{\left(\frac{5}{2} \right)}}} \right)} \]Solve y₀e^(kt) = 2000 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(8) = 1250 \approx 1250,\quad t = \ln{\left(10^{\frac{4}{\ln{\left(\frac{5}{2} \right)}}} \right)} \approx 10.05 \)
Lines: 2 proved, 2 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 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ 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 model evaluated, and the time found by a numerical root-finder |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (misleading) — Step 1 incorrectly refers to a 'doubling time' when the quantity grows from 200 to 500 (a factor of 2.5, not 2). Step 4 presents an unnecessarily complex and non-standard form for the time t, obscuring the simple logarithmic solution.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (misleading) 2026-10-09 — Step 1 incorrectly refers to a 'doubling time' when the quantity grows from 200 to 500 (a factor of 2.5, not 2). Step 4 presents an unnecessarily complex and non-standard form for the time t, obscuring the simple logarithmic solution.gpt-oss:20b: fail (error) 2026-10-09 — The solution misapplies the exponential model: the constant k is correctly found, but the calculation for y(8) is omitted and the expression for the time when the quantity reaches 2000 is incorrect and confusing. The final formula for t does not follow from the model and would mislead a student.qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly sets up the exponential growth model, determines the rate constant k, and calculates the required values. The final expression for t is algebraically equivalent to the simplified form 4*ln(10)/ln(5/2).
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/exponential_models, checked 2026-10-09 with SymPy 1.14.0.