Answer to Question #124162 in Calculus for joe chegg

Question

Suppose a particle P is moving in the plane so that its coordinates are given by P(x, y),
where x = 4 cos 2t, y = 7 sin 2t.
(i) By finding a, b ∈ R such that x^2/a^2+y^2/b^2= 1, show that P is travelling on an elliptical path.
(ii) Let L(t) be the distance from P to the origin. Obtain an expression for L(t).
(iii) How fast is the distance between P and the origin changing when t = π/8?

Answer 1

As $\sin^22t+\cos^22t=1$ hence

(i) $\frac{x^2}{4^2}+\frac{y^2}{7^2}=1$ is ellipse,where $a=4,b=7$

(ii) $L(t)=\sqrt{16\cos^22t+49\sin^22t}$

(iii) $L'(t)=\frac{66\cos2t\sin2t}{\sqrt{16\cos^22t+49\sin^22t}}$ then $L'(\pi/8)\approx5.8$ is the rate of change of distance between P and the origin