Net change and accumulation from a rate
Problem 4.345 · medium
Oil leaks into a containment boom at \( \displaystyle r(t) = 100 e^{- \frac{t}{2}} \) gallons per hour, starting from 50 gallons. How much has collected at \( \displaystyle t = 2 \) hours?
- The net change over [0, T] is the integral of the rate (the Fundamental Theorem of Calculus).Reviewed
- \[ \int\limits_{0}^{2} 100 e^{- \frac{t}{2}}\, dt = 200 - \frac{200}{e} \]The change over the interval.✓ Proved
- \[ 250 - \frac{200}{e} \]Add the starting amount.✓ Proved
Answer \( 250 - \frac{200}{e} \approx 176.4\ \text{gal} \)
✓ 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 by integrating the rate function and adding the initial condition. The algebraic steps are verified and the final result matches the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the Fundamental Theorem of Calculus to find the accumulated amount by integrating the rate function and adding the initial condition. The algebraic steps are verified and the final result matches the stated answer.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the Fundamental Theorem of Calculus to find the net change and adds the initial condition to find the total amount. The algebraic steps are correct.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.