# Minimizing distance word problem

• Mar 11th 2010, 02:59 PM
hustleboyy
Minimizing distance word problem
a train leaves the station at 10:00 pm and travels due north at a speed of 100km/h. another train has been heading due west at 120km/h and reaches the same station at 11:00pm. at what time were the two trains cloet together?

no clue how to do it
so different form textbook examples
• Mar 12th 2010, 08:31 AM
hollywood
Obviously, it will be at 10:00 or some time after, since train 1 is at the station and train 2 is coming directly toward the station. And it will be 11:00 or some time before, since after that, train 1 is going away from the station to the north and train 2 is going away from the station to the west, so they are only getting further apart.

If you define t to be the number of hours after 10:00, the position of train 1 is (0,100t) and the position of train 2 is (120(1-t),0). So you can use the distance formula $\displaystyle d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ to get the distance between the two trains as a function of t. Then you find the minimum by taking the derivative and setting it to zero.

As a check, you can use some intuition - at 10:30, the trains are the same distance from the station, but train 2 is moving faster so they're still getting closer together. So your answer should be a little after 10:30.

Post again if you're still having trouble.