Maximizing Volume

[newageslacke]

I am doing max/min problems in calculus. The problem is to maximize the volume of a rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid x^2/4 + y^2 + z^2/9 = 1

[timo]

I´d try this: There is only one degree of freedom you have: The extension of the box in one of the axis-direction (because this extension automatically gives you the other two extensions, at least when you assume that the box touches the ellipsiod, which seems a sensible assumption). Try to express the volume of the box by this one parameter, then find the maximum as usual (by taking the derivative with respect to this parameter). I cannot guarantee you that it works because I haven´t worked it out, but it´s what would be my first attempt.

[Rocket Man]

you have to specify at least two axis values for the other to fall into place.

if you specify one, you end up with an ellipse of possible solutions.

 

i think substitution is the way to go, im not sure how far you can take it, but i've taken it down to v = f(x,y)

potentially you could make a 3d graph and find the maximum value for the volume by taking lots of x,y combinations.

a little programming wouldn't go astray.

[newageslacke]

I solved the problem, thanks for the help. I ended up using Lagrange Multipliers.