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Find the volume of the largest rectangular box with edges parallel to the axes that can...

Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid

x^2/9+y^2/36+z^2/1=1

Hint: By symmetry, you can restrict your attention to the first octant (where x,y,z≥0), and assume your volume has the form V=8xyz. Then arguing by symmetry, you need only look for points which achieve the maximum which lie in the first octant. Maximum volume:

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