ship problem: at what time are they closest to each other?

[dopey9]

A ship P is travelling due East at 30 km/h and a Ship Q is travelling due South at 40 km/h.
Both ships keep constant speed and course. At t=0 they are each 10 km from the point of intersection of their courses and moving towards the point.

Iv found the co-ordinates of Q relative to P at t=0
---->X=(10,10)km

iv also found the velocity of Q relative to P----> V= V[Q] -V[P]
-----.V= (0,40) - V(30,0)
----- V= (-30,40)km/h

but im struggling to find the time at which P and Q are closest to each other...is there anyone that can help??? thankz

 

[skeeter]

start by designating an origin ... let (0,0) be the point of intersection of their respective course lines.

at t = 0, ship P is at (-10,0)
ship P's position (in km) from the origin as functions of time in hours is
x = 30t - 10
y = 0

at t = 0, ship Q is at (0,10)
ship Q's position (in km) from the origin as functions of time in hours is
x = 0
y = -40t + 10

the distance between the two ships at any time t would be ...

D = sqrt[(30t - 10)² + (-40t + 10)²]

or

D² = (30t - 10)² + (-40t + 10)²

let D² = Z ... minimize Z w/r to t

Z = (30t - 10)² + (-40t + 10)²

dZ/dt = 60(30t - 10) - 80(-40t + 10)

set dZ/dt = 0, t = 7/25 hr ... at that time, the ships will be a minimum distance of
2 km apart