Ostal regulations specify that a parcel sent by parcel post may have a combined length and girth (distance around) of no more than 126 inches. if a rectangular package has a square end, find the maximum volume of this package.

Let x represent the edge length of the square end. Then the area of the end of the package is x², and the length of the package is 126 - 4x. The overall volume is ...

V = x² * (126 - 4x)

V = - 4x³ + 126x²

The volume will be a maximum where the derivative of volume with respect to x is zero.

V' = - 12x² + 252x

V' = - 12x (x - 21)

This will be zero for x = 0 and for x = 21

The maximum volume of the package is

V = (21 in) ² (126 in - 4*21 in)

V = 18,522 in³