A robotic welding arm follows a path C C\,C in a horizontal workspace, defined by the parametric equations
x=12t2+2,y=2t−8t,t>0 x = \frac{1}{2}t^2 + 2, \quad y = 2t - \frac{8}{t}, \quad t > 0 x=21t2+2,y=2t−t8,t>0where x x\,x and y y\,y are coordinates in centimetres. The path C C\,C intersects the xxx-axis at the point QQQ.
Determine the xxx-coordinate of QQQ.
A safety barrier is represented by the line lll, which is the normal to the path C C\,C at the point PPP. Given that t=4t = 4t=4 at PPP:
Write down the coordinates of PPP.
Using calculus, show that an equation of l l\,l is
8x+5y=110 8x + 5y = 110 8x+5y=110The region R R\,R is bounded by the path C C\,C from Q Q\,Q to PPP, the line l l\,l from P P\,P to the xxx-axis, and the xxx-axis between Q Q\,Q and the line intercept.
Using algebraic integration, find the exact volume of the solid of revolution formed when the region R R\,R is rotated through 2π 2\pi\,2π radians about the xxx-axis.
Practise Edexcel A Level Maths Differentiation with exam-style questions for A Level Maths. 327 questions covering 12.1 Gradients of Curves, 12.2 Differentiation from first principles, 12.3 Differentiating x^n, 12.4 Differentiating Quadratics, 12.5 Differentiating functions with two or more terms, 12.6 Gradients, Tangents and Normals, 12.7 Increasing and Decreasing Functions, 12.8 Second Order Derivatives, 12.9 Stationary Points, 12.10 Sketching Gradient Functions, and 12.11 Modelling with Differentiation, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
