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

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

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

Boat A's position is given by x_a(t)=3-t, y_a(t)=2t-4x_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)=3-2tx_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?