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suppose a straight river 0.5km wide flows due east at a constant speed 3km/h.Suppose a boat row at 2km/h from a point A on the south shore to point B,which is k km downstream from the point directly north of point A of the other shore.What direction should you head to cross the river.

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I assume you want to wind up at the point $B$ and that the direction of travel is constant.

Let $\theta$ be the angle at which the boat travels with respect to the south shore.

The velocity vector of the boat is ${\bf v}=(3+ 2\cos\theta) {\bf i}+ 2(\sin\theta ) {\bf j}$.

To find $\theta$, solve the system of equations: $$ (3+ 2\cos\theta)t = k,\quad \text {and}\quad 2(\sin\theta )t =\textstyle{1\over2} $$ where $t$ is the time it takes for the boat to reach point $B$.

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