1. ## Legrange Volume problem

Heres the problem:

Find the maximum possible volume of a rectangular box that has its base in the xy-plane and its upper vertices on the elliptic paraboloid z = 9 - x^2 - y^2

So we are trying to maximize V = xyz.

I need 2 constraints. One is: z = 9 - x^2 - y^2

Can someone help me find the other? I know it has to do with base in xy plane but I cant derive an equation out fo that.

2. Originally Posted by messianic
Heres the problem:

Find the maximum possible volume of a rectangular box that has its base in the xy-plane and its upper vertices on the elliptic paraboloid z = 9 - x^2 - y^2

So we are trying to maximize V = xyz.

I need 2 constraints. One is: z = 9 - x^2 - y^2

Can someone help me find the other? I know it has to do with base in xy plane but I cant derive an equation out fo that.
Since the paraboloid is radially symmetric about the z-axis, the only boxes resting on the xy-plane that have all four upper vertices on the paraboloid are those whose bases are centered on the origin of the xy-plane; and by the radial symmetry, you can assume the base is oriented so that its sides meet the x and y axes at right angles. Hence you want to maximize (2x)(2y)z = 4xyz = V subject to the constraint that x^2 + y^2 + z = 9, with x and y > 0. The answer I'm getting for the volume is 81/8.