Two ships pursue straight-line courses at steady speeds. At each time t (in hours, where 0 less-than-or-equal t less-than-or-equal 14), the positions of the two ships are given by:

Question

Two ships pursue straight-line courses at steady speeds. At each time t (in hours, where 0 less-than-or-equal t less-than-or-equal 14),  the positions of the two ships are given by:
Ship A (10t+35, 20t+25)
Ship B (11t+15, 22t+15)
These coordinates are with reference to a particular coordinate system, with distance measured in kilometres.

What is the closest distance to which the ships approach each other and when does this closest distance occur? Give you answers exactly.

Steve · Answer

Clearly the distance z between the ships can be found using z^2 = (Ax-Bx)^2 + (Ay-By)^2 = (-t+20)^2 + (-2t+10)^2 = 5t^2 - 80t + 500 2z dz/dt = 10t-80 dz/dt = 0 when t=8. So, just evaluate z at t=8.