99 ’s family is using a shipping trailer that is 5 feet tall, 4 feet wide, and 12 feet long to load boxes of apples in. each box the apples are packed in has to have a volume of 5184 in3. design a box that will fill the trailer with no wasted space, and meet the volume requirements. show work

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.

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.

a box 3 3/4 feet long or 4+ feet.

step-by-step explanation: