Answer:
102.6 miles
Step-by-step explanation:
In this problem, the initial distance between the two trains is the distance between New York and Boston, so
d = 240 miles
We know that the average speed of train A is 60 mph, and that train A travels from Boston to New York, so we can write its velocity as
[tex]v_A=+60 mph[/tex]
At the same time, train B travels from New York to Boston, so its velocity will be negative (since it is going in the opposite direction).
Also, we know that the speed of train B is 3/4 that of train A, so we have:
[tex]v_B = -\frac{3}{4}v_A=-\frac{3}{4}(60) = -45 mph[/tex]
So, velocity of train B is 45 mph towards Boston.
Taking Boston as zero position as reference, x = 0, the position of train A at time t can be written as
[tex]x_A(t)=v_At = +60 t[/tex]
While the position of train B at time t is
[tex]x_B(t)=d-v_B t = d-45 t[/tex]
The two trains meet when they have same position, so when
[tex]x_A=x_B[/tex]
And so when
[tex]60t=d-45t[/tex]
Solving for t,
[tex]60t+45t=d\\105t=240\\t=\frac{240}{105}=2.29 h[/tex]
The position at which they meet is
[tex]x_A(2.29)=60\cdot 2.29=137.4 mi[/tex]
However, this is the distance from Boston; so the distance from New York is:
[tex]d=240-137.4=102.6 mi[/tex]
