Solutions start on the next page: consider a rectangular box with dimensions x, 2x and y. Find the dimensions which maximize the volume of this box subject to the constraint that the diagonal of the box is 1 meter: consider a rectangular box with dimensions 4x, x and y. What dimensions maximize volume subject to the constraint that the dimensions of the box sum to 3 meters: consider a rectangular box with dimensions x, 2x and y. Find the dimensions which maximize the volume of this box subject to the constraint that the diagonal of the box is 1 meter. If d is the diagonal of the box, then d satis es the equation d2 = x2 + (2x)2 + y2. Since the diagonal must be 1 meter, this becomes. Di erentiating both of these equations implicitly with respect to x yields.