Two autonomous vehicles move in a plane, with position in kilometres after t hours given by rA(t)=(2t,1+t),rB(t)=(12−t,7+0.5t),t≥0. Determine when they are closest together and find their minimum separation, giving answers to three decimal places where appropriate.
The separation vector is s=rB−rA=(12−3t,6−0.5t). Thus d2=(12−3t)2+(6−0.5t)2=437t2−78t+180. Differentiating and setting the derivative to zero gives 237t−78=0⇒t=37156≈4.216h. The quadratic coefficient is positive, so this is the minimum. Substitution gives dmin2=37576,dmin=3724≈3.946km.
Forms the separation vector.
[1 mark]
Forms the distance-squared function.
[2 marks]
Determines the minimising time.
[2 marks]
Determines the minimum separation.
[1 mark]
Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.