Respuesta :
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] hours
It is going to take 1.4 hours for the two motorcycles to meet.
