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Find the height, volume, and both radii of the elliptical cylinder of the largest volume that can fit within the ellipsoid (x^2)/(a^2) + (y^2)/(b^2) + (z^2)/(c^2) = 1, where a, b, and c are the distances from the origin to the edge of the ellipsoid along the x, y, and z axes respectively.
What did you try so far? Where did you see problems? How does your sketch look like?
Did you find any relations between the different parameters of the cylinder? Hint: I would try to express the radii as function of the height. And the volume as function of the radii and the height. If you have that, it is a simple exercise to find the maximal volume, which depends on one parameter only.