Net change and accumulation from a rate
Problem 4.966 · easy
Oil leaks into a containment boom at \( \displaystyle r(t) = 2 t + 5 \) gallons per hour, starting from 50 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).
- \[ \int\limits_{0}^{5} \left(2 t + 5\right)\, dt = 50 \]The change over the interval.✓ Proved
- \[ 100 \]Add the starting amount.✓ Proved
Answer \( 100 \approx 100\ \text{gal} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly calculates the definite integral of the rate function as 50, when it is actually 62.5. Consequently, the final total of 100 is incorrect; the correct total is 112.5.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly calculates the definite integral of the rate function as 50, when it is actually 62.5. Consequently, the final total of 100 is incorrect; the correct total is 112.5.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The integral of the rate function from 0 to 5 is 60, not 50. The solution incorrectly calculates the definite integral, leading to a wrong final answer of 100 instead of 110.gpt-oss:20b: pass 2026-10-12
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 2026-10-12 with SymPy 1.14.0.