There are parallel train tracks running from Abilene to Imogene. The tracks run in a straight line between the two cities. On a particular morning, Train A begins the journey from Abilene to Imogene on the first track. At the exact same time, Train B begins the journey from Imogene to Abilene on the second track. The two trains always travel at constant velocities.
The two trains pass each other at a small town called Xanadu. From there, Train A takes another 5 hours to reach Imogene, while Train B takes another 10 hours to reach Abilene.
The next day, Train A embarks on the return trip to Abilene at the exact same time Train B leaves for Imogene. This time, they pass each other at a small town called Yonkers, whereby Train A takes another 5 hours to reach Abilene while Train B takes another 10 hours to reach Imogene.
If the distance between Yonkers and Xanadu is 500 miles, then how fast is each train traveling?
A : Abilene
I : Imogene
X : Xanadu
Y : Yonkers
t : Time to reach X = Time to reach Y
v(A) : Velocity of train A
v(B) : Velocity of train B
AX = t*v(A) = 10*v(B)
XI = t*v(B) = 5*v(A)
Solving these gives
v(A) = v(B)*sqrt(2)
t = 5*sqrt(2)
Since the description of the trips is the same,
we must conclude that AY = XI.
AX = AY + YX = XI + YX
or
t*v(A) = 5*v(A) + 500
or
5*sqrt(2)*v(A) = 5*v(A) + 500
or
v(A) = 100*(1+sqrt(2)) ~= 241.42 miles/hour
and
v(B) = 50*(2+sqrt(2)) ~= 170.71 miles/hour

Posted by Bractals
on 20070505 11:26:22 