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Home » \( 90 + 7.20 = 97.20 \) - Groen Casting

\( 90 + 7.20 = 97.20 \) - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 After 12 days, the total displacement is \( 7 \times 12 = 84 \equiv 0 \pmod{12} \), returning to the starting segment.

📌 Therefore, the participant returns to segment 1 after:

📌 Question:** An environmental scientist models soil compaction using the vector field \( \vec{F}(x, y) = \langle -y, x \rangle \), representing rotational stress. Find the work done by this field along the circular path \( \vec{r}(t) = \langle 3\cos t, 3\sin t \rangle \) for \( t \in [0, 2\pi] \).

📌 The work is given by the line integral:

📌 W = \int_C \vec{F} \cdot d\vec{r}

📌 Let \( \vec{r}(t) = \langle 3\cos t, 3\sin t \rangle \), so:

📌 \vec{r}'(t) = \langle -3\sin t, 3\cos t \rangle, \quad t \in [0, 2\pi]

📌 Now, evaluate \( \vec{F}(\vec{r}(t)) = \langle -3\sin t, 3\cos t \rangle \)

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