A man launches his boat from point A on a bank of a straight

A man launches his boat from point A on a bank of a straight river, 3 km wide, and wants to reach point B, 3 km downstream on the opposite bank, as quickly as possible. 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, how many kilometers downstream from C should he aim for with his boat? (Assume that the speed of current is negligible).

Solution

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C ...X ...B

we shall solve in general form & then substitute numerical values

let width of river AC = w,
landing point X be x km from C
& let the angle downstream (CAX) be z
it can be shown that if running speed / rowing speed = k,
then sin z = w/k will give the minimum time, which yields
x = w/(k^2-1)

plugging in,
x = 3/((8/6)^2 - 1) = 9/7 km > 3 km
WHICH MEANS HE SHOULD ROW DIRECTLY TO B !!