A man can row a boat with $4kmph$ in still water. If he is crossing a river where the current is $8kmph$, in what direction (w.r.t the river flow) should he head the boat to cross the river in shortest path? In what time he will cross? (Width of river is $8km$)

To cross by the shortest path, the boat's upstream velocity component must cancel the river current, giving $4 \sin \theta = 2$ where $\theta$ is the angle with the perpendicular, meaning he heads at $120^\circ$ with the river flow. His velocity component across the river is $4 \cos 30^\circ = 2\sqrt{3}$ km/h. The time to cross the $8$ km wide river is $8 / (2\sqrt{3}) = 4 / \sqrt{3}$ hours. The result is $120^\circ, \cfrac { 4 }{ \sqrt { 3 } } hr$.

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