Thread: Calculus optimization

A train leaves the station at 10:00 and travels due north at a speed of 60 km/h. Another train has been heading due west at 45 km/h and reaches the same station at 11:00. At what time were the two trains closest together?

An actual train queastion. And here I thought they were just a tv trope.

Train 1 distance from station is 60t Where t=0 at 10
Train 2 distance from station is 45-45t
Pythagorean theorem gives us the distance between trains:
sqrt ((60t)^2 + (45-45t)^2)
The square root has a minimum at the same time as its contents so we can ignore its leaves us with finding the minimum of
5625t^2 - 4050T +2025
Differentiating to find the minimum gives us
11250t - 4050 = 0 or t=.36
Multiply by 60 to get 21.6 minutes after 10