Basic questions about finding a volume formula for a box inside of an ellipsoid,I want to maximize the volume of a box, with sides parallel to the xy, xz and yz-planes, with the box inside of an ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 = 1So, my answer, using Lagrange multipliers, turns out to be wrong, at least comparing my work to the solution given.I had thought to maximize "length times height times width", so I figured that the objective function should be f ( x , y , z ) = x y x z y z = x 2 y 2 z 2 .The solution instead maximizes a different function: f ( x , y , z ) = 2 x 2 y 2 zI am guessing that the solution is indeed correct? And the point I probably missed was this: the ellipsoid is centered at 0. So, sketching out the box on paper, "length times height times width" does look like 2 x 2 y 2 z.What do you think?Also, how does an equation of an ellipsoid not centered at 0 look like? Would it be something like this: ( x − 1 ) 2 a 2 + ( y − 2 ) 2 b 2 + ( z − 3 ) 2 c 2 = 1 ?would this be an ellipsoid centered at (1,2,3)?

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Basic questions about finding a volume formula for a box inside of an ellipsoid,I want to maximize the volume of a box, with sides parallel to the xy, xz and yz-planes, with the box inside of an ellipsoid            x      2              a      2        +            y      2              b      2        +            z      2              c      2        =  1So, my answer, using Lagrange multipliers, turns out to be wrong, at least comparing my work to the solution given.I had thought to maximize &quot;length times height times width&quot;, so I figured that the objective function should be   f  (  x  ,  y  ,  z  )  =  x  y  x  z  y  z  =      x    2        y    2        z    2  .The solution instead maximizes a different function:  f  (  x  ,  y  ,  z  )  =  2  x  2  y  2  zI am guessing that the solution is indeed correct? And the point I probably missed was this: the ellipsoid is centered at 0. So, sketching out the box on paper, &quot;length times height times width&quot; does look like   2  x  2  y  2  z.What do you think?Also, how does an equation of an ellipsoid not centered at 0 look like? Would it be something like this:            (      x      −      1              )        2                    a      2        +            (      y      −      2              )        2                    b      2        +            (      z      −      3              )        2                    c      2        =  1  ?would this be an ellipsoid centered at (1,2,3)?

Yaretzi Melendez · Accepted Answer

Step 1The correct equation for the ellispoid centered at the origin is            x      2              a      2        +            y      2              b      2        +            z      2              c      2        =  1.It is obvious that the resulting (optimal) box would be centered at the origin.Step 2If we denote one of the vertices of the box by (x, y, z), then the sides have lengths 2x, 2y, 2z assuming   x  ,  y  ,  z  &amp;gt;  0. With this, the volume is 8xyz.

torfuqx · Accepted Answer

Explanation:A dimension check would’ve told you that your expression for the volume of the inscribed box was incorrect. A volume should have a dimension of       length    3  , but       x    2        y    2        z    2   has a dimension of       length    6  .

I want to maximize the volume of a box, with sides parallel to the xy, xz and yz-planes, with the box inside of an ellipsoid x^2/a^2+y^2/b^2+z^2/c^2=1

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