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.
717 exam-style questions on Edexcel A Level Maths Differentiation, covering 9.1 Differentiating sin x and cos x, 9.2 Differentiating exponentials and logarithms, 9.3 The Chain Rule, 9.4 The Product Rule, 9.5 The Quotient Rule, 9.6 Differentiating Trigonometric Functions, 9.7 Parametric Differentiation, 9.8 Implicit Differentiation, 9.9 Using Second Derivatives, and 9.10 Rates of Change. Each one has a worked solution and a mark scheme showing where the marks go.