Exponential growth and decay
Problem 5.356 · medium
A quantity grows exponentially: it is 100 at \( \displaystyle t = 0 \) and 150 at \( \displaystyle t = 2 \). Find it at \( \displaystyle t = 4 \), and when it reaches 1000.
- y = y₀e^{kt}; the given doubling time, half-life or second data point fixes k.
- k = log(3/2)/2.
- \[ 225 \]y(4).✓ Proved
- \[ \frac{2 \ln{\left(10 \right)}}{\ln{\left(\frac{3}{2} \right)}} = \ln{\left(10^{\frac{2}{\ln{\left(\frac{3}{2} \right)}}} \right)} \]Solve y₀e^(kt) = 1000 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(4) = 225 \approx 225,\quad t = \ln{\left(10^{\frac{2}{\ln{\left(\frac{3}{2} \right)}}} \right)} \approx 11.36 \)
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 refers to the second data point as a 'doubling time' in step 1, which is factually wrong (the quantity grows by a factor of 1.5, not 2). Additionally, step 4 presents an algebraic identity rather than the actual solution for t, failing to provide the numerical answer or the simplified exact form requested.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution incorrectly refers to the second data point as a 'doubling time' in step 1, which is factually wrong (the quantity grows by a factor of 1.5, not 2). Additionally, step 4 presents an algebraic identity rather than the actual solution for t, failing to provide the numerical answer or the simplified exact form requested.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly sets up the exponential growth model, determines the rate constant k, and solves for the requested values. The final expression for t is algebraically equivalent to the standard form ln(10)/k.
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 2026-10-07 with SymPy 1.14.0.