(i assume you're in calculus 3, so getting everything in one variable is not one of my objectives).
draw a diagram to see what i am talking about. the vertex that touches the plane will be the vertex that is opposite the z-axis above the xy-plane. from this we extract the following information.
we define the length to be the side of the base along the x-axis, the width the side of the base along the y-axis and the height to be the height of the vertex that touches the plane.
Let the length of the box be
Let the width of the box be
Then the height of the box is .........(i solved for z from the equation of the plane)
Thus, the volume is given by:
now we want to find the maximum of this function. can you continue? i gave you more than a hint