Question

The post office will accept packages whose combined length and girth is at most 42 inches. (the girth is the perimeter/distance around the package perpendicular to the length; for a rectangular box, the length is the largest of the three dimensions.) what is the largest volume that can be sent in a rectangular box? (round answer to 2 decimal places.)

Answer 1

If x represents the length of the box, then 42-x will be the girth. Since the largest area for a given girth is that of a square, the side length of the square cross section is (42 -x)/4.

The volume as a function of package length is then
.. v(x) = x((42-x)/4)^2
This has a maximum at x=14. The corresponding volume is 686 in^3.