Boat A's position is given by #x_a(t)=3-t, y_a(t)=2t-4# and boat B's position is given by #x_b(t)=4-3t, y_b(t)=3-2t#. The distance units are kilometres and the time units are hours. At what time are the boats closest to each other?

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Answer 1 · 2017-06-13T12:18:47

The time is #=3/2h#

I tried it this way :

The position of #A# is #r_A=(3-t,2t-4)#

The position of #B# is #r_B=(4-3t,3-2t)#

The difference of their positions is

#=r_A-r_B=(3-t-4+3t, 2t-4-3+2t)#

#=(-1+2t,4t-7)#

The distance is

#p=sqrt((-1+2t)^2+(4t-7)^2)#

#=sqrt(1-4t+4t^2+49-56t+16t^2)#

#=sqrt(20t^2-60t+50)#

To calculate the closest distance, we calculate the first derivative

#(dp)/dt=1/(2sqrt(20t^2-60t+50))*(40t-60)#

When

#(dp)/dt=0#, #=>#, #40t-60=0#

#t=60/40=3/2h#