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March 21st, 2017, 07:54 PM   #1
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Partial Derivative Application for 3D object

Find the volume of the largest rectangular box with edges parallel to the axes that can be
inscribed in the ellipsoid 2x^2 + 72y^2 + 18z^2 = 288.

Answer: 256/sqrt(3)

I would be much appreciated if anyone kindly point me to the right direction. Thanks!
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March 21st, 2017, 08:33 PM   #2
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well by symmetry your corner points will be

$(\pm a, \pm b, \pm c)$

with volume $V=8abc$

you can reduce this a bit using the formula for the ellipsoid to get points at

$\left(\pm a, \pm b, \pm \sqrt{\dfrac{288 - 2a^2 - 72b^2}{18}}\right)$

with volume

$V=8ab \sqrt{\dfrac{288 - 2a^2 - 72b^2}{18}}$

As usual take the gradient of this expression, with respect to $a,~b$ and set it equal to the vector 0 and solve for $a,~b$

Solving this might be a bit of a mess but Mathematica returns a pretty digestible answer so it might not be that bad.

The way to ensure that this extreme points is a maximum is a bit more complicated for a 3D than a 2D case.

Do you know how to use the two dimensional 2nd derivative test to do this?

Last edited by romsek; March 21st, 2017 at 09:13 PM.
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