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 mys, we can use a quadratic function as a basic model for the rate of water flow x units from the west bank: f (x) = 3/400 x(40 - x).
(a) A boat proceeds at a constant speed of 5 m/s from a point A on the west bank while maintaining a heading perpendicular to the bank. How far down the river on the opposite bank will the boat touch shore? Graph the path of the boat.
(b) Suppose we would like to pilot the boat to land at the point B on the east bank directly opposite A. If we maintain a constant speed of 5 mys and a constant heading, find the angle at which the boat should head. Then graph the actual path the boat follows. Does the path seem realistic?