Divergent paths? Two boats leave a port at the same lime, one traveling west at 20 mi/hr and the other traveling southwest at 15 mi/hr. At what rate is the distance between them changing 30 min after they leave the port?

Solution: Step1 To find At what rate is the distance between them changing 30 min after they leave the port Step2 Given that Two boats leave a port at the same lime, one traveling west at 20 mi/hr and the other traveling southwest at 15 mi/hr Step3 Step4 The boat is traveling west at 20mi/hr.. x1 = - (20 mi/hr) t ; y1 = 0 are coordinates of boat #1 at time t in hours We have, dx1/dt = - 20; dy1/dt = 0 Other boat is traveling southwest at 15 mi/hr Then, x2 = - (15 mi/hr) [cos (45 degrees)] t ; y2 = - (15 mi/hr) [sin (45 degrees)are coordinates of boat #2 at time t in hours dx/dt = - 10.6065; dy2/dt = - 10.6065 Distance between two boats D = [(y2 y1) 2 + (x2 x1) 2] We want to know dD/dt = when t = 0.5 hours ˆ ˆ D = [(10.6065 t 0) 2 + (10.6065 t 20 t) 2] = [112.5 t 2 + 88.2 t 2] D = 14.2 t Differentiate both side we get dD/dt = 14.2 mi/hr Therefore,At 14.2 mi/hr rate is the distance between them changing 30 min after they leave the port.