Putting z in terms of x and y

1. Oct 19, 2004

circa415

Minimizing Volume

If I have a linear plane that cuts through the first octant so that there is a z, y, and x intercept so that you have a triangular face in the octant, is there any way I can put z in terms of x and y? Here's what I'm trying to do

I'm trying to find some linear plane that cuts off the smallest volume in the first octant. The plane must pass through a specific point (let's say it's (2,3,4). I figured since V = Bh and h will stay the same (distance from origin to the point), you will want to minimize the area of the triangular face. So I set each vertice of the triangle as (x,0,0), (0,y,0), and (0,0,z) and take half the cross product of the two vectors <x,0,-z> and <0,y,-z>. But I only know how to find the minimum for only two variables. Is there anyway I could put z in terms of x and y?

I know there is the eq. ax+by+cz+d=0 but I don't see how it would relate to what I'm trying to do right now (finding the mimimum area of the triangular face) and if there is a connection, I don't see it. I think I just use it later on to find the equation for the plane, but right now I'm trying to find the x,y, and z intercepts of the plane..

Or am I just approaching this problem in the wrong way?

Thanks for the help

Last edited: Oct 19, 2004