Net change and accumulation from a rate
Problem 4.665 · easy
Water flows into a tank at a rate of \( \displaystyle r(t) = 3 t^{2} + 2 \) liters per minute. The tank holds 50 liters at \( \displaystyle t = 0 \). How much water is in the tank at \( \displaystyle t = 6 \)?
- The net change over [0, T] is the integral of the rate (the Fundamental Theorem of Calculus).Reviewed
- \[ \int\limits_{0}^{6} \left(3 t^{2} + 2\right)\, dt = 228 \]The change over the interval.✓ Proved
- \[ 278 \]Add the starting amount.✓ Proved
Answer \( 278 \approx 278\ \text{L} \)
✓ 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 condition to find the final amount. The logic is sound and the calculations are correct.
Every verdict on record (4)
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 the initial condition to find the final amount. The logic is sound and 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 condition. The arithmetic 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.