Apackage in the shape of a rectangular box can be mailed by the us postal service if the sum of its length and girth (the perimeter of a cross-section perpendicular to the length) is at most 108 in. find the dimensions of the package with largest volume that can be mailed. (for full cr

For a given girth, the largest area will be that of a square. Then for some length x, the volume will be
.. V = x((108 -x)/4)^2
.. = (1/16)*x*(108 -x)^2
The largest volume will be found where the derivative of this with respect to x is zero.
.. V' = (1/16)*((108 -x)^2 -2x(108 -x))
.. 0 = (108 -x)(108 -x -2x)
.. x = 108 or 108/3 = 36

The dimensions of the largest package that can be mailed are
.. 36 inches in length and 72 inches in girth

The volume of that package is 11,664 in^3.