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What is the maximum volume of a box that can be placed inside an ellipsoid $\frac{x^2}{16}+\frac{y^2}{9}+\frac{z^2}{25}=1$

The volume of a box is $V=xyz$ so I need to find $x,y,z$ with respect to the ellipsoid conditions and then find the maximum point.

$x=\sqrt{16-\frac{16y^2}{9}-\frac{16z^2}{25}}$

$y=\sqrt{9-\frac{9x^2}{16}-\frac{9z^2}{25}}$

$z=\sqrt{25-\frac{25x^2}{16}-\frac{25y^2}{9}}$

We take only the positive root as the length but on the other hand a negative root is just a length to the other direction and if 2 of the variables are negative the volume will be still a positive number

Martin Sleziak
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gbox
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  • Looks like a job for Lagrange multipliers. What have you tried so far? – amd Apr 01 '16 at 07:43
  • Just a sec, I have no info in the box so its volume is $xyz$. – gbox Apr 01 '16 at 07:45
  • @amd I tried to find z and find the maximum point – gbox Apr 01 '16 at 07:45
  • You’re assuming that the sides of the inscribed box are parallel to the coordinate axes. That’s not explicitly stated in your question. With that assumption, $xyz$ is only the volume of the part of the box in the 1st octant. – amd Apr 01 '16 at 08:16
  • Anyway, your approach will work. If you know about Lagrange multipliers (as described in Matthias’ answer), try that instead as the computations involved will be much simpler. – amd Apr 01 '16 at 08:20
  • @amd so should o take both positive and negative roots? For x,y,z? – gbox Apr 01 '16 at 08:32
  • See what they are first. By symmetry, I’d expect them to come in $\pm$ pairs. – amd Apr 01 '16 at 08:34
  • If you look at the related question (in the sidebar), you can find some similar questions, like this one or this one. – Martin Sleziak Apr 01 '16 at 09:55
  • @MartinSleziak I looked at them but all of them use Lagrange multipliers – gbox Apr 01 '16 at 09:56
  • @gbox Not true. There are some answers based on AM-GM inequality. (And I am not sure what is wrong with using Lagrange multipliers.) – Martin Sleziak Apr 01 '16 at 10:10

1 Answers1

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One simple standard procedure would be when you have the condition:

$$f(x,y,z)=\frac{x^2}{16}+\frac{y^2}{9}+\frac{z^2}{25}=1$$

and you want to maximize

$$V(x,y,z)=xyz$$

to set $$\vec\nabla f(x,y,z)=\lambda\cdot\vec\nabla V(x,y,z)$$

their gradient looking in the same direction. After that you have to choose the maxima out of the extrema.

This is the geometric approach learned in my analyses course.

Matthias
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