The airline policy states that all baggage must be box-shaped and the sum of length, width and height should not exceed 108 in.

The dimensions of a square based box with greatest volume has to be determined. Let the length and width of the box be X, the height of the box is 108 - 2*X.

The volume of the box is `V = X*X*(108 - 2*X) = 108X^2 - 2X^3`

Solve `(dV)/(dX) = 0` to determine the required value of X

`(dV)/(dX) = 216X - 6X^2`

`216X - 6X^2 = 0`

=> X = 36

The dimensions of the box with maximum volume is 36x36x36