Exponential growth and decay
Problem 5.283 · medium
A quantity grows exponentially: it is 200 at \( \displaystyle t = 0 \) and 300 at \( \displaystyle t = 5 \). Find it at \( \displaystyle t = 10 \), and when it reaches 2000.
- y = y₀e^{kt}; the given doubling time, half-life or second data point fixes k.
- k = log(3/2)/5.
- \[ 450 \]y(10).✓ Proved
- \[ \frac{5 \ln{\left(10 \right)}}{\ln{\left(\frac{3}{2} \right)}} = \ln{\left(10^{\frac{5}{\ln{\left(\frac{3}{2} \right)}}} \right)} \]Solve y₀e^(kt) = 2000 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(10) = 450 \approx 450,\quad t = \ln{\left(10^{\frac{5}{\ln{\left(\frac{3}{2} \right)}}} \right)} \approx 28.39 \)
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 (error) — The solution incorrectly labels the given data point as a 'doubling time' or 'half-life' in step 1, which is factually wrong (the quantity grows from 200 to 300, not doubling). Additionally, step 4 presents an identity rather than the solution for t, failing to explicitly state the final answer t ≈ 28.39 derived from the equation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution incorrectly labels the given data point as a 'doubling time' or 'half-life' in step 1, which is factually wrong (the quantity grows from 200 to 300, not doubling). Additionally, step 4 presents an identity rather than the solution for t, failing to explicitly state the final answer t ≈ 28.39 derived from the equation.gpt-oss:20b: fail (style) 2026-10-06 — The last sentence is unclear and does not explicitly state the time t; it mixes an identity with the desired result, which could confuse a student.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly sets up the exponential growth model, determines the rate constant k, and calculates the values for t=10 and the time to reach 2000. The algebraic forms are correct.
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 2026-10-06 with SymPy 1.14.0.