Question

the parametric equations and parameter intervals for the motion of a particle in the xy - plane are given below. identify the particles path by finding a cartesian equation for it. graph the cartesian equation. indicate the portion of the graph traced by the particle and the direction of motion.

x = 2cos(2t), y = 2sin(2t), 0 ≤ t ≤ π

the cartesian equation for the particle is x² + y² =

Explanation:

Step1: Express \(\cos(2t)\) and \(\sin(2t)\) in terms of \(x\) and \(y\)

Given \(x = 2\cos(2t)\), then \(\cos(2t)=\frac{x}{2}\). Given \(y = 2\sin(2t)\), then \(\sin(2t)=\frac{y}{2}\).

Step2: Use the Pythagorean identity \(\cos^{2}\alpha+\sin^{2}\alpha = 1\)

Substitute \(\alpha = 2t\), \(\cos(2t)=\frac{x}{2}\) and \(\sin(2t)=\frac{y}{2}\) into the identity \(\cos^{2}(2t)+\sin^{2}(2t)=1\).
We get \((\frac{x}{2})^{2}+(\frac{y}{2})^{2}=1\).

Step3: Simplify the equation

Expand \((\frac{x}{2})^{2}+(\frac{y}{2})^{2}=1\) to \(\frac{x^{2}}{4}+\frac{y^{2}}{4}=1\), and then multiply through by \(4\) to obtain \(x^{2}+y^{2}=4\).

When \(t = 0\), \(x=2\cos(0)=2\), \(y = 2\sin(0)=0\). When \(t=\frac{\pi}{4}\), \(x=2\cos(\frac{\pi}{2}) = 0\), \(y=2\sin(\frac{\pi}{2})=2\). When \(t=\frac{\pi}{2}\), \(x=2\cos(\pi)=- 2\), \(y=2\sin(\pi)=0\). As \(t\) increases from \(0\) to \(\pi\), the particle moves counter - clockwise around the circle \(x^{2}+y^{2}=4\) starting from the point \((2,0)\) and making a full - circle (since when \(t = 0\), \((x,y)=(2,0)\) and when \(t=\pi\), \(x = 2\cos(2\pi)=2\), \(y=2\sin(2\pi)=0\))

Answer:

The Cartesian equation for the particle is \(x^{2}+y^{2}=4\). The particle moves counter - clockwise around the circle \(x^{2}+y^{2}=4\) starting from the point \((2,0)\) and making a full - circle as \(t\) goes from \(0\) to \(\pi\).