Answer:
A box 1' by 1.5' by 2' will meet the requirements
Step-by-step explanation:
The desired volume is (5184 in^3)/(1728 in^3/ft^) = 3 ft^3. Clearly, a 1 ft by 1 ft by 3 ft box will fit in the trailer. (The 3 ft dimension would be stacked along the 12 ft dimension of the trailer.) The 5' x 4' x 12' = 240 ft^2 trailer will hold 80 boxes.
That long, skinny box can be made a little more cubical by cutting the 3' dimension in half and doubling one of the 1' dimensions.
Then, in the 5' by 4' by 12' trailer, the boxes would have corresponding dimensions of 1' by 2' by 1.5'. This would allow them to be packed 5 high by 2 wide by 8 deep (along the 12' length).
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<em>Development of alternate sets of dimensions</em>
If we restrict ourselves to dimensions having an integer numbers of inches, we find that the greatest common denominator of 5 feet and 5184 cubic inches is 12 inches. That is, 12 inches is the tallest the box can be and still have an integer number of square inches of area. If the box is 12" high, there will be 5 layers, and each layer of boxes will be 80/5 = 16 boxes.
The arrangement of boxes on the floor of the trailer can be 1x16, 2x8, or 4x4 boxes in each layer. The first number could correspond to the number of boxes along the length of the trailer, and the second number the width, or vice versa.
For example, 1x16 could mean each box is 12 ft long and 4'/16 = 3 inches wide. Or, it could mean each box is 4' long and 12'/16 = 9 inches wide. The latter seems a more practical size than the former.
Likewise, a 2x8 arrangement could be of boxes 6 ft long and 6 inches wide, or 2 ft long and 18 inches wide. (These are the dimensions reported above.)
Finally, a 4x4 arrangement would be of boxes 1 ft wide and 3 ft long.
Of all these sets of dimensions, the one we like the best is 1 ft high, 2 ft wide, and 18 inches long.
