At noon, ship a is 180 km west of shipb. ship a is sailing east at 35 km/h and ship b is sailing north at 30 km/h. how fast is the distance between the ships changing at 4:00 pm?

Accepted Solution

If we let the coordinates of each ship be (x, y) with the positive directions of these coordinates corresponding to East and North, then the positions of the ships t hours after noon are

Ship A
  (-180+35t, 0)

Ship B
  (0, 30t)

The distance between them is the "Pythagorean sum" of the difference in their coordinates:
  d = √((-180 +35t)² +(-30t)²)
    = √(32400 -12600t +2125t²)
The rate of change of this distance is
  dd/dt = (2125t -6300)/√(32400 -12600t +2125t²)

At 4 pm, the value of this rate of change is
  (2125*4 -6300)/√(32400 -12600*4 +2125*4²)
  = 2200/√16000
  ≈ 17.39 km/h

The distance between the ships is increasing at about 17.39 km/h at 4 pm.