Question

A train leaves the station at 10:00 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 11:00. At what time were the 2 trains closests?

Answer 1

after t hours,
north train has gone 100t from the station
west train is 120-120t km from the station
so the distance z between the trains is
z^2 = (100t)^2 + (120-120t)^2
= 24400t^2 - 28800t + 14400
= 400(61t^2 - 72t + 36)
z dz/dt = 800(61t-36)
dz/dt = 0 at t = 36/61 = 35.4 minutes
so at 10:35:25 the trains were closest

Answer 2

say time t = hours after 10 am
north train at y = 100 t
west train x= -120 (t-1)
z = d^2 = x^2+ y^2= (100t)^2 + 120^2 (t^2 - 2 t + 1)
z = 10,000 t^2 + 14,400 t^2 - 28,000 t + 14,000
z = 24, 400 t^2 - 28,000 t +14,000
dz / dt = 48,800 t - 28,000
min or max when dz/dt = 0
t = 28.0/48.8 = 0.574 hr = 34.4 minutes after 10am