Home›Calculus 1›Net change and accumulation from a rate›Problem 4.820

Net change and accumulation from a rate

Problem 4.820 · easy

A particle starts at position 100 m with velocity \( \displaystyle v(t) = t^{2} + 9 \) m/s. Where is it at \( \displaystyle t = 6 \)?

The net change over [0, T] is the integral of the rate (the Fundamental Theorem of Calculus).
\[ \int\limits_{0}^{6} \left(t^{2} + 9\right)\, dt = 126 \]
The change over the interval.✓ Proved
\[ 226 \]
Add the starting amount.✓ Proved

Answer \( 226 \approx 226\ \text{m} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The integral is computed from t=0 to t=6, but the problem states the particle starts at position 100 m (presumably at t=0). The solution correctly adds 100 to the change, but the first sentence claims the net change is the integral over [0, T] without explicitly linking the start time t=0 to the initial condition, which is a minor logical gap, but more importantly, the phrasing 'The net change over [0, T]' is vague. However, the real issue is that the solution does not explicitly state that the initial position is at t=0. If the particle started at t=100m at some other time, the integral bounds would be wrong. Assuming standard convention that 'starts at' implies t=0, the math is correct. But wait, looking closer: The solution calculates the integral from 0 to 6. It adds 100. The result is 226. The math is correct. Is there an error? The sentence says 'The net change over [0, T] is the integral...'. It doesn't specify T=6 in the sentence, but the equation does. This is acceptable. Let's look for a deeper error. Ah, the problem says 'starts at position 100 m'. It does not explicitly say 'at t=0'. However, in calculus problems, 'starts' usually implies t=0. If we assume t=0, the solution is correct. Is there a misleading part? No. Is there a style issue? The sentence is a bit generic. Let's re-read the prompt constraints. 'Check that the setup models the problem as stated'. The setup assumes t=0 is the start time. This is a standard assumption. Is it an error? No. Is it misleading? No. Wait, let me check the integral value. Integral of t^2+9 from 0 to 6 is [t^3/3 + 9t] from 0 to 6 = 6^3/3 + 9*6 = 216/3 + 54 = 72 + 54 = 126. 100 + 126 = 226. The math is correct. Why would I fail it? Maybe the sentence is too vague? 'The net change over [0, T]'. It should say 'The displacement from t=0 to t=6'. But [0, T] with T=6 is implied. Let's look at the 'Stated answer'. 226. The solution derives 226. It seems correct. Let me look for a subtle error. 'A particle starts at position 100 m'. Does it start at t=0? Usually yes. If it started at t=1, the answer would be different. Without explicit 'at t=0', is it an error to assume t=0? In introductory calculus, 'starts' implies t=0. I will pass it unless there is a clear error. Wait, looking at the JSON output format. If it's correct, I say pass. Is it correct? Yes. Is the sentence misleading? No. Is it style? Maybe. But the prompt asks to find what is wrong. If nothing is wrong, pass. Let me double check the 'proved' lines. Line 2: Integral(t**2 + 9, (t, 0, 6)) = 126. Correct. Line 3: 226 = 226. Correct. Sentence 1: 'The net change over [0, T] is the integral of the rate'. This is a statement of FTC. It is correct. I will pass.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-10 — The integral is computed from t=0 to t=6, but the problem states the particle starts at position 100 m (presumably at t=0). The solution correctly adds 100 to the change, but the first sentence claims the net change is the integral over [0, T] without explicitly linking the start time t=0 to the initial condition, which is a minor logical gap, but more importantly, the phrasing 'The net change over [0, T]' is vague. However, the real issue is that the solution does not explicitly state that the initial position is at t=0. If the particle started at t=100m at some other time, the integral bounds would be wrong. Assuming standard convention that 'starts at' implies t=0, the math is correct. But wait, looking closer: The solution calculates the integral from 0 to 6. It adds 100. The result is 226. The math is correct. Is there an error? The sentence says 'The net change over [0, T] is the integral...'. It doesn't specify T=6 in the sentence, but the equation does. This is acceptable. Let's look for a deeper error. Ah, the problem says 'starts at position 100 m'. It does not explicitly say 'at t=0'. However, in calculus problems, 'starts' usually implies t=0. If we assume t=0, the solution is correct. Is there a misleading part? No. Is there a style issue? The sentence is a bit generic. Let's re-read the prompt constraints. 'Check that the setup models the problem as stated'. The setup assumes t=0 is the start time. This is a standard assumption. Is it an error? No. Is it misleading? No. Wait, let me check the integral value. Integral of t^2+9 from 0 to 6 is [t^3/3 + 9t] from 0 to 6 = 6^3/3 + 9*6 = 216/3 + 54 = 72 + 54 = 126. 100 + 126 = 226. The math is correct. Why would I fail it? Maybe the sentence is too vague? 'The net change over [0, T]'. It should say 'The displacement from t=0 to t=6'. But [0, T] with T=6 is implied. Let's look at the 'Stated answer'. 226. The solution derives 226. It seems correct. Let me look for a subtle error. 'A particle starts at position 100 m'. Does it start at t=0? Usually yes. If it started at t=1, the answer would be different. Without explicit 'at t=0', is it an error to assume t=0? In introductory calculus, 'starts' implies t=0. I will pass it unless there is a clear error. Wait, looking at the JSON output format. If it's correct, I say pass. Is it correct? Yes. Is the sentence misleading? No. Is it style? Maybe. But the prompt asks to find what is wrong. If nothing is wrong, pass. Let me double check the 'proved' lines. Line 2: Integral(t**2 + 9, (t, 0, 6)) = 126. Correct. Line 3: 226 = 226. Correct. Sentence 1: 'The net change over [0, T] is the integral of the rate'. This is a statement of FTC. It is correct. I will pass.
gpt-oss:20b: pass 2026-10-10
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies the Fundamental Theorem of Calculus to find the displacement and adds it to the initial position. 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 2026-10-10 with SymPy 1.14.0.