Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 10 - Parametric Equations and Polar Coordinates - 10.1 Exercises - Page 666: 21

Answer

The particle with position (x,y) moves 3 times clockwise along an ellipse. The equation of this ellipse is: $\frac 1 {25}x^2 + \frac{1}{4}y^2= 1$ It starts and ends at $(0,-2)$

Work Step by Step

1. Convert the equation to a Cartesian one, so we can identify the type of curve: $x = 5sin (t)$ $x^2 = 25sin^2(t)$ $\frac{1}{25}x^2 = sin^2(t)$ $y = 2cos(t)$ $y^2 = 4cos^2(t)$ $\frac{1}{4}y^2 = cos^2(t)$ Adding the equations: $\frac 1 {25}x^2 + \frac{1}{4}y^2= sin^2(t) + cos^2(t)$ $\frac 1 {25}x^2 + \frac{1}{4}y^2= 1$ - As we can see by the equation, this curve is an ellipse with a radius of $\sqrt {25} = 5$ along the x-axis, and a radius of $\sqrt 4 = 2$ along the y-axis. And with its center at $(0,0)$ 2. Find the initial and final point: Initial point: $t = -\pi$ $x = 5 sin(-\pi) = 5(0) = 0$ $y = 2cos(-\pi) = 2(-1) = -2$ Final point: $t = 5 \pi$ $x = 5sin(5\pi) = 5(0) = 0$ $y = 2cos(5 \pi) = 2(-1) = -2$ - Since $sin(t)$ determines the $x$ position, and $cos(t)$ determines the $y$ position, the particle will move clockwise along the ellipse. - For simple trigonometric functions like $sin(t)$ and $cos(t)$, the particle represented by $(x,y)$ completes a rotation after $t$ increasing by $2\pi$. From $-\pi$ to $5\pi$, there is a total of $6\pi$. Therefore, the particle does a total of 3 complete rotations along the elipse.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.