Net change and accumulation from a rate
Problem 4.964 · easy
Oil leaks into a containment boom at \( \displaystyle r(t) = 2 t^{2} + 4 \) gallons per hour, starting from 1200 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).
- \[ \int\limits_{0}^{2} \left(2 t^{2} + 4\right)\, dt = \frac{40}{3} \]The change over the interval.✓ Proved
- \[ \frac{3640}{3} \]Add the starting amount.✓ Proved
Answer \( \frac{3640}{3} \approx 1213\ \text{gal} \)
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 starting amount is 1200 gallons, but the solution adds 3640/3 (which is 1200 + 40/3) to the integral result, effectively double-counting the initial amount. The correct final amount should be 1200 + 40/3 = 3640/3, but the logic in step 3 implies adding 3640/3 to the change, or simply states the final answer without showing the addition of 1200 to 40/3 clearly. Wait, 3640/3 IS 1200 + 40/3. Let's re-read carefully. Step 3 says '3640/3 = 3640/3 -- Add the starting amount.' This is a tautology that doesn't show the addition. It implies the final answer is 3640/3. Is 3640/3 correct? 1200 + 40/3 = 3600/3 + 40/3 = 3640/3. The final number is correct. However, the sentence 'Add the starting amount' is attached to a line that just equates the final answer to itself. It fails to model the operation 'Initial + Change = Final'. It presents the result without the derivation step, which is misleading about how the result was obtained. But is it an error? The prompt asks to check if sentences are correct. The sentence 'Add the starting amount' is an instruction or description of a step that isn't shown in the equation. The equation is a tautology. This is misleading because it hides the actual calculation (1200 + 40/3). A student might think 3640/3 is just a magic number. However, looking closer at the provided solution text: '3. [equation, proved] 3640/3 = 3640/3 -- Add the starting amount.' This is logically vacuous. It does not show 1200 + 40/3 = 3640/3. It is misleading.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The starting amount is 1200 gallons, but the solution adds 3640/3 (which is 1200 + 40/3) to the integral result, effectively double-counting the initial amount. The correct final amount should be 1200 + 40/3 = 3640/3, but the logic in step 3 implies adding 3640/3 to the change, or simply states the final answer without showing the addition of 1200 to 40/3 clearly. Wait, 3640/3 IS 1200 + 40/3. Let's re-read carefully. Step 3 says '3640/3 = 3640/3 -- Add the starting amount.' This is a tautology that doesn't show the addition. It implies the final answer is 3640/3. Is 3640/3 correct? 1200 + 40/3 = 3600/3 + 40/3 = 3640/3. The final number is correct. However, the sentence 'Add the starting amount' is attached to a line that just equates the final answer to itself. It fails to model the operation 'Initial + Change = Final'. It presents the result without the derivation step, which is misleading about how the result was obtained. But is it an error? The prompt asks to check if sentences are correct. The sentence 'Add the starting amount' is an instruction or description of a step that isn't shown in the equation. The equation is a tautology. This is misleading because it hides the actual calculation (1200 + 40/3). A student might think 3640/3 is just a magic number. However, looking closer at the provided solution text: '3. [equation, proved] 3640/3 = 3640/3 -- Add the starting amount.' This is logically vacuous. It does not show 1200 + 40/3 = 3640/3. It is misleading.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The initial amount is stated as 1200 gallons, but the solution adds 3640/3 (which is approximately 1213.33) instead of 1200. The final answer should be 1200 + 40/3 = 3640/3, but the intermediate step implies adding the final answer to itself or misinterpreting the initial value. Specifically, 1200 + 40/3 = 3600/3 + 40/3 = 3640/3. The equation '3640/3 = 3640/3' is a tautology that hides the error in the setup: it claims the result of adding the starting amount is 3640/3, but doesn't show the addition of 1200. Wait, 1200 + 40/3 IS 3640/3. So the math is correct. Let me re-read carefully. Problem: Start 1200. Rate 2t^2+4. Time 0 to 2. Integral of 2t^2+4 from 0 to 2 is [2/3 t^3 + 4t] from 0 to 2 = 2/3(8) + 8 = 16/3 + 24/3 = 40/3. Total = 1200 + 40/3 = 3600/3 + 40/3 = 3640/3. The solution says: 1. Net change is integral. 2. Integral is 40/3. 3. 3640/3 = 3640/3 -- Add the starting amount. Step 3 is logically incoherent as a derivation. It states the final answer equals itself, labeling it 'Add the starting amount'. It fails to show the addition of 1200. A student reading this would not know where 3640/3 came from relative to 1200 and 40/3. It skips the actual arithmetic step: 1200 + 40/3 = 3640/3. Instead, it just asserts the final number. This is misleading because it doesn't demonstrate the operation performed.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.