A man launches his boat from point A on a bank of a straight river, 4 km wide, a

Question

A man launches his boat from point A on a bank of a straight river, 4 km wide, and wants to reach point B, 2 km downstream on the opposite bank, as quickly as possible (see the figure below). He could row his boat directly across the river to point C and then run to B, or he could row directly to B, or he could row to some point D between C and B and then run to B. If he can row 6 km/h and run 8 km/h, where should he land to reach B as soon as possible? (We assume that the speed of the water is negligible compared to the speed at which the man rows.) ______ km from C

Explanation / Answer

Basically, imagine point D is of horizontal distance k from point A such that when k=0, he travels to point C and when k=1, he travels directly to A. We can then also imagine the distances from A to D as x and the distance from D to B to be y. We know that he can row across the river at a rate of 8km/h and can run at a rate of 6km/h. We may express this in the form of two equations: dx/dt=6 dy/dt=8 Hence: x=6t, t=x/6 y=8t, t=y/8 If the total time is the time to travel x and the time to travel y, then the total time can be understood to be: T=x/6+y/8 Now we both make these functions of the distance k. We can see, from the diagram, that k is the difference between y and 1. We also can see that k forms a pythagorean triangle with distance x. Hence, we may express this as two seperate relationships: y=1-k x=sqrt(4+k^2) This means we may express our former relationship of total time with respect to the distance k. We get: T=(1/6)(sqrt(4+k^2))+(1/8)(1-k) where 0

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