Exponential growth and decay
Problem 5.291 · medium
A quantity grows exponentially: it is 200 at \( \displaystyle t = 0 \) and 400 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.Reviewed
- k = log(2)/4.Reviewed
- \[ 800 \]y(8).✓ Proved
- \[ \frac{4 \ln{\left(10 \right)}}{\ln{\left(2 \right)}} = \ln{\left(10^{\frac{4}{\ln{\left(2 \right)}}} \right)} \]Solve y₀e^(kt) = 2000 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(8) = 800 \approx 800,\quad t = \ln{\left(10^{\frac{4}{\ln{\left(2 \right)}}} \right)} \approx 13.29 \)
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the exponential growth model, calculates the rate constant k based on the doubling time, and derives the correct values for y(8) and the time t when y=2000. The algebraic forms provided in the equations are correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the exponential growth model, calculates the rate constant k based on the doubling time, and derives the correct values for y(8) and the time t when y=2000. The algebraic forms provided in the equations are correct.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the exponential growth model, calculates the growth constant k based on the doubling time, and applies the formula to find the value at t=8 and the time to reach 2000. The algebraic steps are correct and the final approximations are accurate.
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.