Question 18 – 9.7 Parametric Differentiation - Questions

A parabolic acoustic mirror for a long-range microphone is modeled by the parametric equations $5t^2 \text{ and } y = 10t, \quad -3 \le t \le 3$

Determine a Cartesian equation for the profile of the mirror in the form $y^2 = f(x)$ .

A sound sensor is located at point $B (5, 0)$ . A specific point $A$ on the mirror's profile corresponds to the parameter $t = a$ , where $a > 1$ . The tangent to the mirror at $A$ makes an angle $θ\theta$ with a line through $A$ parallel to the $x$ -axis. The line segment $A B$ makes an angle $ϕ\phi$ with the positive $x$ -axis.

(i) By finding an expression for $dydx\frac{dy}{dx}$ in terms of $t$ , show that $tan⁡θ=1a\tan \theta = \frac{1}{a}$ .

(ii) Find $tan⁡ϕ\tan \phi$ in terms of $a$ .

(iii) Hence, prove that $tan⁡2θ=tan⁡ϕ\tan 2\theta = \tan \phi$ .

A parabolic acoustic mirror for a long-range microphone is modeled by the parametric equations $5t^2 \text{ and } y = 10t, \quad -3 \le t \le 3$

Determine a Cartesian equation for the profile of the mirror in the form $y^2 = f(x)$ .

(i) By finding an expression for $dydx\frac{dy}{dx}$ in terms of $t$ , show that $tan⁡θ=1a\tan \theta = \frac{1}{a}$ .

(ii) Find $tan⁡ϕ\tan \phi$ in terms of $a$ .

(iii) Hence, prove that $tan⁡2θ=tan⁡ϕ\tan 2\theta = \tan \phi$ .

9.7 Parametric Differentiation Questions

9.7 Parametric Differentiation Questions