your friend left his home 2 hours ago and cycles due north at 30 km/h. you have been cycling due west at 20 km/h and arrives at his home now. At what time were the two of you closest to each other?

Answer 1

at time t hours, the distance z is

z^2 = (40-20t)^2 + (30t)^2

now just find when dz/dt = 0

Answer 2

the friend's distance north is 30 t

your distance east is ... 40 - 20 t

it's a right triangle

use Pythagoras to find the hypotenuse
... which is the distance

find the minimum - with respect to t