Net change and accumulation from a rate
Problem 4.824 · easy
A town's population grows at \( \displaystyle r(t) = - t^{2} + 6 t + 30 \) people per year. The population is 0 at \( \displaystyle t = 0 \). What is it at \( \displaystyle t = 6 \)?
- The net change over [0, T] is the integral of the rate (the Fundamental Theorem of Calculus).Reviewed
- \[ \int\limits_{0}^{6} \left(- t^{2} + 6 t + 30\right)\, dt = 216 \]The change over the interval.✓ Proved
- \[ 216 \]Add the starting amount.✓ Proved
Answer \( 216 \approx 216\ \text{people} \)
✓ Nihil obstat Lines: 2 proved, 1 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ 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 | numerical quadrature of the rate |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the Fundamental Theorem of Calculus to find the net change and adds the initial population. The logic is sound and the steps are clearly justified.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies the Fundamental Theorem of Calculus to find the net change and adds the initial population. The logic is sound and the steps are clearly justified.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies the Net Change Theorem. The integral calculation is correct, and the logic of adding the initial population (0) to the net change is sound.
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/net_change, checked 2026-10-10 with SymPy 1.14.0.