Find the maximum volume for a rectangular solid in the first octant with one vertex at the origin and opposite vertex on the plane x + y + z = 1.

Question

I have to find the maximum volume for a rectangular solid in the first octant ( $x \ge 0 , y \ge 0 , z \ge 0$ ) with one vertex at the origin and opposite vertex on the plane $x + y + z = 1$ .

I need to use the method of Lagrange multipliers to find a set of equations that when solved will give the max volume of the rectangle solid

Answer 1

A point on the plane $x+y+z=1$ has coordinates $(u, v, 1 - u - v)$.

If one vertex of the cuboid is $(0,0,0)$ then the volume is

$W=uv(1-u-v)$

set partial derivatives $W_u=0;\;W_v=0$

that is $$\begin{cases} u (1-u-v)-u v=0\\ v (1-u-v)-u v=0\\ \end{cases} $$