Two objects, A and B, are moving along two different straight lines at constant speeds. With reference to a particular coordinate system in which distance is measured in metres, the position of A at...

Two objects, A and B, are moving along two different straight lines at constant speeds. With reference to a particular coordinate system in which distance is measured in metres, the position of A at time t (in minutes) is (3t−2,−2t+4), and the position of B is (−t−3,t−4).


A student attempts to find the shortest distance between A and B as shown below. There are two lines where the working does not follow on from the previous line.

   d^2 =25t^2 −40t+65
= 25 (t^2 − 8/5 t) + 65
= 25 (t − 4/5t)^2 − 16/25 + 65
= 25 (t − 4/5t)^2 + 1609/25

The minimum value of d^2 occurs when t = 45 . Hence the minimum distance is 64 m (to 2 s.f.).

-  Find and describe the two mistakes.
-  Write out a correct solution, stating when the minimum value of d2
occurs and the shortest distance between A and B.