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),0leq tleqpi )
the cartesian equation for the particle is ( x^{2}+y^{2}=4 )
choose the correct graph that represents this motion

Explanation:

Step1: Analyze the parametric equations

We have \(x = 2\cos(2t)\) and \(y = 2\sin(2t)\).
Using the trigonometric identity \(\cos^{2}\theta+\sin^{2}\theta = 1\). Here \(\theta=2t\), \(\cos(2t)=\frac{x}{2}\) and \(\sin(2t)=\frac{y}{2}\).

Step2: Substitute into the identity

Substituting into \(\cos^{2}(2t)+\sin^{2}(2t)=1\), we get \((\frac{x}{2})^{2}+(\frac{y}{2})^{2}=1\), which simplifies to \(x^{2}+y^{2}=4\). This is the equation of a circle centered at the origin \((0,0)\) with radius \(r = 2\).
For the direction of motion:
When \(t = 0\), \(x=2\cos(0)=2\), \(y = 2\sin(0)=0\).
When \(t=\frac{\pi}{4}\), \(x=2\cos(\frac{\pi}{2})=0\), \(y=2\sin(\frac{\pi}{2})=2\).
As \(t\) increases from \(0\) to \(\frac{\pi}{2}\), the point \((x,y)\) moves from \((2,0)\) to \((0,2)\) in the counter - clockwise direction.

Answer:

The Cartesian equation is \(x^{2}+y^{2}=4\). The graph is a circle of radius \(2\) centered at the origin with counter - clockwise motion. (Assuming one of the options (if we had full visual details) that shows a circle \(x^{2}+y^{2}=4\) with a starting point at \((2,0)\) and moving counter - clockwise)