Exponential growth and decay
Problem 5.495 · medium
A quantity grows exponentially: it is 200 at \( \displaystyle t = 0 \) and 500 at \( \displaystyle t = 2 \). Find it at \( \displaystyle t = 4 \), and when it reaches 2000.
- y = y₀e^{kt}; the given doubling time, half-life or second data point fixes k.
- k = log(5/2)/2.
- \[ 1250 \]y(4).✓ Proved
- \[ \frac{2 \ln{\left(10 \right)}}{\ln{\left(\frac{5}{2} \right)}} = \ln{\left(10^{\frac{2}{\ln{\left(\frac{5}{2} \right)}}} \right)} \]Solve y₀e^(kt) = 2000 for t: t = ln(M/y₀)/k.✓ Proved
Answer \( y(4) = 1250 \approx 1250,\quad t = \ln{\left(10^{\frac{2}{\ln{\left(\frac{5}{2} \right)}}} \right)} \approx 5.026 \)
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: fail (error) — The solution incorrectly computes y(4). Using y(4)=200e^{k·4}=200e^{2k}=200·(5/2)=500, not 1250. The t‑value for 2000 is correct, but the earlier numerical error invalidates the solution.qwen3.6:27b-mlx: fail (error) — The solution incorrectly labels the given data point (y(2)=500) as a 'doubling time' or 'half-life' in step 1, which is factually wrong (it is a 2.5x growth). Additionally, step 4 presents an algebraic identity rather than the actual calculation for t, failing to provide the numerical answer requested.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-12 — The solution incorrectly computes y(4). Using y(4)=200e^{k·4}=200e^{2k}=200·(5/2)=500, not 1250. The t‑value for 2000 is correct, but the earlier numerical error invalidates the solution.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly labels the given data point (y(2)=500) as a 'doubling time' or 'half-life' in step 1, which is factually wrong (it is a 2.5x growth). Additionally, step 4 presents an algebraic identity rather than the actual calculation for t, failing to provide the numerical answer requested.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to explicitly calculate the initial value y₀ = 200, which is required to set up the specific exponential model y = 200e^{kt}. Without stating y₀, the derivation of k and subsequent values lacks the necessary context, making the solution incomplete and potentially confusing for a student.
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 2026-10-12 with SymPy 1.14.0.