The vertical cross-section of a custom-built acoustic shell for an outdoor concert stage is being designed. The shell rests on a flat horizontal stage floor. Using a coordinate system where the origin O O\,O is at the left-most point of the shell's base and the xxx-axis lies along the stage, the right-most point of the base is PPP. The shape of the cross-section is modelled by the equation
x2+y2=kx−10y x^2 + y^2 = k\sqrt{x} - 10y x2+y2=kx−10ywhere x x\,x and y y\,y are the horizontal and vertical distances from the origin in metres, and k k\,k is a constant.
The total width of the shell base OP OP\,OP is 16 metres. Determine the value of k k\,k for this design.
Show that the maximum height of the acoustic shell above the stage is approximately 7.1 metres.
State one limitation of using this mathematical equation to model the physical acoustic shell.
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.