Net change and accumulation from a rate
Problem 4.670 · easy
A town's population grows at \( \displaystyle r(t) = t + 11 \) people per year. The population is 50 at \( \displaystyle t = 0 \). What is it at \( \displaystyle t = 4 \)?
- The net change over [0, T] is the integral of the rate (the Fundamental Theorem of Calculus).
- \[ \int\limits_{0}^{4} \left(t + 11\right)\, dt = 52 \]The change over the interval.✓ Proved
- \[ 102 \]Add the starting amount.✓ Proved
Answer \( 102 \approx 102\ \text{people} \)
Lines: 2 proved, 1 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 | ✓ 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: fail (misleading) — The solution skips the explicit addition of the initial population (50) to the computed change (52). The equation 102 = 102 does not show how 102 was obtained, leaving a student confused about the final step.qwen3.6:27b-mlx: pass — The solution correctly applies the Net Change Theorem: final value equals initial value plus the integral of the rate of change. The calculations are correct.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-08 — The solution skips the explicit addition of the initial population (50) to the computed change (52). The equation 102 = 102 does not show how 102 was obtained, leaving a student confused about the final step.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies the Net Change Theorem: final value equals initial value plus the integral of the rate of change. The calculations are correct.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies the Fundamental Theorem of Calculus to find the net change and adds it to the initial population. The logic is sound and the result is 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/net_change, checked 2026-10-08 with SymPy 1.14.0.