Chapter 10.7, Problem 40AYU

In Problems 39-42, find parametric equations for an object that moves along the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ with the motion described.

The motion begins at $\left(0,3\right)$, is counterclockwise, and requires 1 seconds for a complete revolution.

To determine

To find: The parametric equations for an object that moves along the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ with the motion described.

Answer to Problem 40AYU

$x=-2\text{sin}2\pi t$ and $y=3\text{cos}2\pi t$, $0\le t\le 1$

Explanation of Solution

Given:

The motion begins at $\left(0,3\right)$ is counterclockwise, and requires 1 second for a complete revolution.

Calculation:

The motion begins at $\left(0,3\right)$, then $x=0$ and $y=3$ when $t=0$.

From the equation $\frac{x^2}{4}+\frac{y^2}{9}=1,a=\pm 2$ and $b=\pm 3$.

The motion is counter clockwise, we must have $x$ negative and $y$ positive initially.

Hence, $x=-2\text{sin}wt$ and $y=3\text{cos}wt$.

$2\pi /w=1$, $w=\text{ 2}\pi$

Therefore, $x=-2\text{sin}2\pi t$ and $y=3\text{cos}2\pi t$, $0\le t\le 1$.