Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 12 - Vectors and the Geometry of Space - 12.6 Cylinders and Quadric Surfaces - 12.6 Exercises: 22

Answer

$IV$

Work Step by Step

The equation $9x^{2}+4y^{2}+z^{2}=1$ is an ellipsoid so, it can be either Figure $IV$ or Figure $VII$. If we write the given equation in the standard form of an ellipsoid centered at the origin, we will have $\frac{x^{2}}{(1/3)^{2}}+\frac{4y^{2}}{(1/2)^{2}}+\frac{z^{2}}{(1)^{2}}=1$ The numbers $1/3,1/2$ and $1$ are the number of units you go along either side of the $x-, y-$ and $z$-axes, respectively, from the origin, that is, the center of the ellipsoid, to get to the surface of the ellipsoid. Since, we can visualize from the Figure $IV$, has the ellipsoid more elongated along the $z$-axis and 1 (the number under $z^{2})$ is greater than $1/2$ and $1/3$, so it must be Figure $IV$.
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