According to u. s. postal regulations, the girth plus the length of a parcel sent by mail may not exceed 108 inches, where by "girth" we mean the perimeter of the smallest end. what is the largest possible volume of a rectangular parcel with a square end that can be sent by mail? such a package is shown below. assume y>x. what are the dimensions of the package of largest volume?

Let x represent the side length of the square end, and let d represent the dimension that is the sum of length and girth. Then the volume V is given by

V = x² (d - 4x)

Volume will be maximized when the derivative of V is zero.

dV/dx = 0 = - 12x² + 2dx

0 = - 2x (6x - d)

This has solutions

x = 0, x = d/6

a) The largest possible volume is

(d/6) ² (d - 4d/6) = 2 (d/6) ³

= 2 (108 in/6) ³ = 11,664 in³

b) The dimensions of the package with largest volume are

d/6 = 18 inches square by

d - 4d/6 = d/3 = 36 inches long