A cubic foot has a volume of (12 in)³ = 1728 in³. Your apple box is required to have a volume of (5184 in³)/(1728 in³/ft³) = 3 ft³.
The shipping trailer dimensions are not multiples of 3 ft, except in the length direction. The easiest box to design is one that is 1 ft square and 3 ft long.
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For there to be no wasted space, the number of boxes must be ...
... (5 ft)·(4 ft)·(12 ft)/(3 ft³/box) = 240/3 boxes = 80 boxes.
In order to pack the trailer with no wasted space, there must be an integral number of boxes in each direction. There are 43 sets of dimensions that will give a 3 ft³ box that will pack the trailer. The one that has dimensions closest to a cube is
... 24 in wide × 15 in high × 14.4 in deep.
10 of these boxes can be put in the length of the trailer, and they can be stacked 2 wide and 4 high.
