The cross-section of a curved acoustic reflector is modelled by the curve C C\,C with parametric equations
x=−4cos2θ,h=5cosθ,0≤θ≤π2 x = -4 \cos 2\theta, \quad h = 5 \cos \theta, \quad 0 \le \theta \le \frac{\pi}{2} x=−4cos2θ,h=5cosθ,0≤θ≤2πwhere x x\,x is the horizontal displacement in metres from a central axis and h h\,h is the height in metres. The region R R\,R is bounded by the curve CCC, the xxx-axis, and the hhh-axis for the portion of the curve where x≥0x \ge 0x≥0.
(i) Show, making your working clear, that the area of R=∫π/4π/280cos2θsinθ dθR = \int_{\pi/4}^{\pi/2} 80 \cos^2 \theta \sin \theta \, d\thetaR=∫π/4π/280cos2θsinθdθ.
(ii) Hence find, by algebraic integration, the exact value of the area of RRR.
Show that all points on C C\,C satisfy h=ax+bh = \sqrt{ax+b}h=ax+b, where a a\,a and b b\,b are constants to be found.
State the range of the function f(x)=ax+bf(x) = \sqrt{ax+b}f(x)=ax+b for the domain −4≤x≤4-4 \le x \le 4−4≤x≤4.
Practise AQA A Level Maths 1.6 C: Coordinate geometry in the (x, y) plane with exam-style questions for A Level Maths. 110 questions covering 1.6.1 Equation of a straight line, 1.6.2 Coordinate geometry of the circle, 1.6.3 Parametric equations of curves (A-level only), and 1.6.4 Parametric equations in modelling (A-level only), matched to the AQA A Level Maths (7357) 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.
