A package 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.