Question

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?

Answer 1

11,664 inches³

Explanation:

Let 'S' be the length of the side of the square base and 'H' be height of the parcel.

The U.S postal regulations specify that:

`4S +H = 108\\H=108-4S`

The volume of the parcel is given by:

`V=H*S^2\\V=(108-4S)*S^2\\V=108S^2-4S^3`

The value of 'S' for which the derivate of the volume function equals zero yields the maximum volume:

`(dV(S))/(dS)= 216S-12S^2=0\\(216S-12S^2)/(S)=(0)/(S) \\216 - 12S =0\\S=18\ inches`

Solving for 'H':

`H=108-4S = 108-(4*18)\\H= 36\ inches`

The maximum volume is:

`V_(max) = 36*18^2\\V_(max) = 11,664\ inches^3`