Water traveling along a straight portion of a river normally flows fastest in th

Question

Water traveling along a straight portion of a river normally flows fastest in the middle, and the speed slows to almost zero at the banks. Consider a long straight stretch of river flowing north, with parallel banks 40 m apart. If the maximum water speed is 3 m/s, we can use the sine function,

f(x) = 3 sin(?x/40),

as a basic model for the rate of water flow x units from the west bank. Suppose a boater would like to pilot the boat to land at the point B on the east bank directly opposite point A. If the boat maintains a constant heading and a constant speed of 5 m/s, determine the angle at which the boat should head. (Round your answer to one decimal place.)

Explanation / Answer

Try this. Integrate (your sine function minus K) where K is a constant to be solved for. Set this equal to zero and solve for K. The K is the parallel component (parallel to river flow). You have a right triangle with hypotenuse of 5, and one leg is K. You can now find the angle.

I continue on. The integral is equal to [-K*x - (120/pi)*cos(x*40/pi)] evaluate from x = 0 to x=40.

This is: -40*K - (120/pi)*(-1) - (0 -120/pi) = 0. So K = pi/6 So the angle theta is Acos((pi/6) / 5) = 67.54

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