Since Naomi cannot place a partial cube in the box, the maximum number of whole cubes that Naomi can fit in the box is 140 cubes .
The volume of the rectangular box is 22 * 15 * 5 = 1650 cubic inches.
The volume of each cube is 2.25 * 2.25 * 2.25 = 11.71875 cubic inches.
Therefore, the maximum number of cubes that Naomi can fit in the box is 1650 / 11.71875 = 140 cubes.
To find the maximum number of cubes that can fit in the box, we need to find the greatest common divisor (GCD) of the dimensions of the box. The GCD is the largest number that is a divisor of all of the dimensions. The GCD of 22, 15, and 5 is 5. This means that the largest cube that can fit perfectly in the box has a side length of 5 inches.
To find the maximum number of cubes that can fit in the box, we simply divide the volume of the box by the volume of each cube.
1650 / 11.71875 = 140. Therefore, the maximum number of cubes that Naomi can fit in the box is 140.