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.