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.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.