Question

A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 114 inches (see figure). Find the dimensions of the package of maximum volume that can be sent. (Assume the cross section is square.)

Answer 1

From the given figure

Let x represent the cross section of the package

Let y represent the length of the package

The formula for the volume, V, of the package (a cubiod) is

`\begin{gathered} V=\text{LBH} \\ L\text{ is the length} \\ B\text{ is the breadth} \\ H\text{ is the height} \end{gathered}`

And

`\begin{gathered} L=y \\ B=x \\ H=x \end{gathered}`

Substitute for the variables into the formula above

`\begin{gathered} V=y* x* x=x^2y \\ V=x^2y \end{gathered}`

For the perimeter of the crossection

`\begin{gathered} P=x+x+x+x+y=4x+y \\ P=4x+y \\ \text{Where P}=114 \\ 114=4x+y \end{gathered}`

Make y the subject

`\begin{gathered} 114=4x+y \\ y=114-4x \end{gathered}`

Substitute for y into the Volume, V.

`\begin{gathered} V=x^2y \\ V=x^2(114-4x) \\ V=114x^2-4x^3 \end{gathered}`

Differentiating the above expression gives

`V^(\prime)=228x-12x^2`

Where V' = 0

`\begin{gathered} 228x-12x^2=0 \\ Divide\text{ both sides by 12} \\ 19x^{}-x^2=0 \\ x(19-x)=0 \\ x=0 \\ 19-x=0 \\ x=19 \\ x=0\text{ or 19} \end{gathered}`

Since x can't be negative, thus, x is 19 inches

Recall that

`\begin{gathered} y=114-4x \\ \text{Where x}=19 \\ y=114-4(19) \\ y=114-76 \\ y=38 \end{gathered}`

Hence,