Describe the region Ω Ω = ( x , y , z ) : 4 &lt; x 2 + y 2 + z 2 &lt; 9

Question

Describe the region   Ω  Ω  =      (    x    ,    y    ,    z    )    :    4    &amp;lt;          x              2              +          y              2              +          z              2              &amp;lt;    9

Cindy Walls · Accepted Answer

since  4  &amp;lt;      x          2        +      y          2        +      z          2        &amp;lt;  9  ⇒      2          2        &amp;lt;      x          2        +      y          2        +      z          2        &amp;lt;      3          2      so the   Ω is a region between two spheres:      x          2        +      y          2        +      z          2        =      2          2       and      x          2        +      y          2        +      z          2        =      3          2

Brylee Shepard · Accepted Answer

This is the equation of a sphere. We can tell because it has the general formula

x^{2} + y^{2} + z^{2} = r^{2}

Centered at origin

(x - x_{0})^{2} + (y - y_{0})^{2} + (z - z_{0})^{2} = r^{2}

general form of a sphere not centered atthe origin where

x_{0}, y_{0}, z_{0}

are the spheres center
The actual shape of this is more of a spherical shell because wecan see that the function is between the radius 2 and 3. We cantell this because of the general formula presented above. For lackof a good graphing program I will not attempt to draw this object.
This is a spherical shell centered at the origin. We cantell this because it meets the criteria of the first equation. The shell is between r=2 and r=3.

Describe the region Omega Omega = (x,y,z):4<x^2 +y^2 +z^2 < 9

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Answer & Explanation

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