Find max volume of a rectangular box within a given ellipsoid

Hello,

We are working on some multi-variable max/min problems in class but this question seems quite complicated. I tried messing around with it for a while but can't see how to get the answer. If anyone can help I would really appreciate it.

The question asks: Find the volume of the largest rectangular box (with faces parallel to the coordinate planes) that can be inscribed inside the ellipsoid

x²/a²+y²/b²+z²/c²=1

Hey lytwynk.

The first thing you should do is find an expression of the volume of the box in terms of the constants a, b, and c.

One suggestion I have is to use the volume integration method to get the "slice" of the volume that is not inside the region of the box and relate that to the constants a, b, and c.

To make it a little clear picture a two-dimensional analog where we inscribe a box in a circle.

Now consider that we are finding the areas to the left and the right of the box in a 2D sense.

The area of the box + area outside box = area of ellipsoid.

If we choose a width say t units and integrate from (-a,-a+t) for f(x) = 2*SQRT([1 - (x/a)^2]*b^2) [We use the 2 to get the area bounded by the lower curve] that gives the area to the right of the box.

By changing limits to (a-t,a) we get the right volume.

You can then obtain the area above and below the box in a similar fashion and relate the volume to this magical t value.

Once this is done you are maximizing area against this t value and this is just a normal optimization.

Now extend this idea to three dimensions by changing areas to volumes and you should have something to go off.

Ok, I will give this a try however in class we are learning about the second derivative test to find critical points and local max and mins for multivariable functions. Do you know any way to relate this method to the question?

Yes there is an exact analogue and its called the Hessian:

Hessian matrix - Wikipedia, the free encyclopedia

What I was trying to do was get x from the ellipse equation. Then I subbed it into the volume equation then differentiated that wrt y and z with the hope of finding a relation between y and z and the other unknowns, where dV/dy=dV/dz=0, ie. a critical point, but it was pretty messy and I could not figure out anything from there. Anyways I will look into your suggestions.
Thanks.