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Brian and Kim are out for a jog in the country. They running east on the same road until Kim notices a parallel road that might provide more shade. She takes a straight side road that heads $60^\circ$ north of east until it intersects with the parallel road. Kim runs east on the parallel road for $1.5\text{ miles}$ then she takes another side road (straight) that heads $30^\circ$ south of east until she is back on the same road as Brian. Brian is jogging at $4$ mph and Kim is jogging at $6$ mph

If the distance between the parallel roads is $\cfrac{1}{2}\text{ mile}$, exactly how far apart are Brian and Kim when Kim returns to the original road and is he behind her or ahead of her?


Brian is ahead $\cfrac{57+8\sqrt3}{18}\text{ miles}$.


Brian is behind $\cfrac{57-8\sqrt3}{18}\text{ miles}$.


Brian is ahead $\cfrac{8\sqrt3+3}{18}\text{ miles}$.


Brian is behind $\cfrac{8\sqrt3-3}{18}\text{ miles}$.


Brian is neither ahead or behind, they meet at $D$.

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