Chapter 10.7, Problem 39AYU

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 $\left(2,0\right)$, is clockwise, and requires 2 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 39AYU

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

Explanation of Solution

Given:

The equation of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ and the motion is described as follows.

The motion begins at $\left(2,0\right)$ is clockwise, and requires 2 seconds for a complete revolution.

Calculation:

The motion begins at $\left(2,0\right)$, then $x=2$ and $y=0$ 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 clockwise, we must have $x$ positive and $y$ negative initially.

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

$2\pi /w=2$, $w=\pi$

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