Two motorcycles leave home at the same time, heading directly toward each other. They are a total of 196 miles apart. The first motorcycle travels at 65 miles per hour. The second motorcycle travels at 75 miles per hour. How long will it take for the two motorcycles to meet? 1.4 hours2.4 hours19.6 hours5.2 hours

Answer:It is going to take 1.4 hours for the two motorcycles to meet.Step-by-step explanation:Understanding the problem:The position of both motorcycles can be modeled by the following first order equation:[tex]S(t) = S(0) + vt[/tex]In which [tex]S(t)[/tex] is the position at the instant t, [tex]S(0)[/tex] is the initial position, v is the speed in miles per hour and t is the time in hours.To solve this problem, we have to model the equation [tex]S_{1}(t)[/tex] for the position of the first motorcycle and [tex]S_{2}(t)[/tex] for the position of the second motorcycle. They are going to meet at the instant t in which[tex]S_{1}(t) = S_{2}(t)[/tex]I will also suppose that the positive direction is the direction from motorcycle 1 to motorcycle 2. One motorcycle moves in the positive direction, other in the negative, since they are heading directly toward each other. For the starting positions, one is at the position 0 and the other at the position 196. This is because they are a total of 196 miles apart.The position of motorcycle 1:I am going to say that motorcycle 1 is at the position 0. So [tex]S(0) = 0[/tex]. The first motorcycle travels at 65 miles per hour. I am going to say that motorcycle 1 travels in the positive direction, so [tex]v = 65[/tex]. So, the equation for the position of motorcycle 1 is:[tex]S_{1}(t) = 65t[/tex]The position of motorcycle 2:Since the motorcycle 1 starts at the position 0, the second motorcycle has to start at the position 196, so [tex]S(0) = 196[/tex]. Since the motorcycle one travels in the positive direction, the second is traveling in the negative direction, at 75 miles per hour, so [tex]v = -75[/tex]. So, the equation for the position of motorcycle 2 is:[tex]S_{2}(t) =196 - 75t[/tex]How long will it take for the two motorcycles to meet? [tex]S_{1}(t) = S_{2}(t)[/tex][tex]65t = 196 - 75t[/tex][tex]140t = 196[/tex][tex]t = \frac{196}{140}[/tex][tex]t = 1.4[/tex] hoursIt is going to take 1.4 hours for the two motorcycles to meet.