Net change and accumulation from a rate
Problem 4.667 · easy
Oil leaks into a containment boom at \( \displaystyle r(t) = t + 6 \) gallons per hour, starting from 0 gallons. How much has collected at \( \displaystyle t = 5 \) hours?
- The net change over [0, T] is the integral of the rate (the Fundamental Theorem of Calculus).Reviewed
- \[ \int\limits_{0}^{5} \left(t + 6\right)\, dt = \frac{85}{2} \]The change over the interval.✓ Proved
- \[ \frac{85}{2} \]Add the starting amount.✓ Proved
Answer \( \frac{85}{2} \approx 42.5\ \text{gal} \)
✓ 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 accumulated amount from the rate function, and correctly handles the initial condition of 0 gallons.
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 accumulated amount from the rate function, and correctly handles the initial condition of 0 gallons.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 amount (0) to get the final total. The logic 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 2026-10-08 with SymPy 1.14.0.