A parabolic acoustic mirror for a long-range microphone is modeled by the parametric equations
x=5t2 and y=10t,−3≤t≤3 x = 5t^2 \text{ and } y = 10t, \quad -3 \le t \le 3 x=5t2 and y=10t,−3≤t≤3Determine a Cartesian equation for the profile of the mirror in the form y2=f(x)y^2 = f(x)y2=f(x).
A sound sensor is located at point B(5,0)B(5, 0)B(5,0). A specific point AAA on the mirror's profile corresponds to the parameter t=at = at=a, where a>1a > 1a>1. The tangent to the mirror at AAA makes an angle θ\thetaθ with a line through AAA parallel to the xxx-axis. The line segment ABABAB makes an angle ϕ\phiϕ with the positive xxx-axis.
(i) By finding an expression for dydx\frac{dy}{dx}dxdy in terms of ttt, show that tanθ=1a\tan \theta = \frac{1}{a}tanθ=a1.
(ii) Find tanϕ\tan \phitanϕ in terms of aaa.
(iii) Hence, prove that tan2θ=tanϕ\tan 2\theta = \tan \phitan2θ=tanϕ.
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.