text: two ships sailing...? at noon ship a is 200km due west of ship b. ship a is sailing east at 32 km/h and ship b is sailing north at 24 km/h. if they maintain these velocities, how fast is the distance between the two ships changing at 6:00 pm? we should begin by expressing the distance d between them in terms of the elasped time t.  (200-32t)^2 + (24t)^2 = d^2  since we are only interested in the rate of change of distance, we don't need to solve for d. we just differentiate the whole equation.  2*(200-32t)(-32) + 2*(24t)(24) = 2d  d/dt (d) = (-6400 + 1024t + 576 t)/d = (1600t - 6400)/d  at 6:00 pm the elasped time t is 6 hours.  the total distance d is sqrt[(200-32*6)^2 + (24*6)^2] d = sqrt[8^2 + 144^2] = sqrt(20800) = 144.22 km  the rate of change is (1600*6-6400)/144.22 = 22.2 km/hr