Question

According to postal regulations, the girth plus the length of a parcel sent by fourth-class mail may not exceed 108 in. What is the largest possible volume of a rectangular parcel with two square sides that can be sent by fourth-class mail?

Answer 1

11,664 in³

Explanation:

For a square side length of x, the girth is 4x and the length of the parcel is then allowed to be up to 108-4x.

The total volume is the product of the edge lengths, so is ...

V = (108 -4x)(x²) = -4x³ +108x²

This will be a maximum where its derivative is zero:

dV/dx = -12x² +216x = 0 = -12x(x -18)

This is zero for x=0 and for x=18.

The maximum volume parcel is 18 in by 18 in by 36 in, and has a volume of ...

(18 in)(18 in)(36 in) = 11,664 in³