Answer:
11,664 inches³
Step-by-step 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:
[tex]4S +H = 108\\H=108-4S[/tex]
The volume of the parcel is given by:
[tex]V=H*S^2\\V=(108-4S)*S^2\\V=108S^2-4S^3[/tex]
The value of 'S' for which the derivate of the volume function equals zero yields the maximum volume:
[tex]\frac{dV(S)}{dS}= 216S-12S^2=0\\\frac{216S-12S^2}{S}=\frac{0}{S} \\216 - 12S =0\\S=18\ inches[/tex]
Solving for 'H':
[tex]H=108-4S = 108-(4*18)\\H= 36\ inches[/tex]
The maximum volume is:
[tex]V_{max} = 36*18^2\\V_{max} = 11,664\ inches^3[/tex]
