The flight path of a surveillance drone in a vertical cross-section is modelled by the parametric equations
x=2t2−16t,y=t3−16t,t∈R x = 2t^2 - 16t, \quad y = t^3 - 16t, \quad t \in \mathbb{R} x=2t2−16t,y=t3−16t,t∈Rwhere x x\,x represents the horizontal displacement from a control tower and y y\,y represents the height relative to a safety baseline. The drone's path crosses the baseline at the origin and at the points A A\,A and BBB, where A A\,A and B B\,B are distinct points.
Find the coordinates of A A\,A and show that B B\,B has coordinates (96,0)(96, 0)(96,0).
Show that the equation of the tangent to the path at B B\,B is
x+y−96=0 x + y - 96 = 0 x+y−96=0The tangent to the path at B B\,B intersects the path again at the point PPP.
Find, using algebraic methods, the horizontal displacement of the drone at point PPP.