Optimization Problem

[Ghost3k]

Suppose postal requirements are that the maximum of the length plus the girth (cross sectional perimeter) of a rectangular package that may be sent is 300 inches. Find the dimensions of the package with square ends whose volume is to be maximum.

Square side :
Length :

I'm confused on this question, could someone please help me on this one with the steps. I cant really picture how this package is supposed to be drawn.

 

[soroban]

Hello, Ghost3k!

Suppose postal requirements are that the maximum of the length plus the girth
(cross-sectional perimeter) of a rectangular package that may be sent is 300 inches.
Find the dimensions of the package with square ends whose volume is to be maximum.

          *---------* -
         /         /| :
        /         / | :
       /         /  | :
      *---------*   | :
      |         |   | :
      |   * - - | - * L
      |  .      |  .| :
      | .       | . | :
      |.        |.  | :
Girth * - - - - *   | :
      |         |   * -
      |         |  /
      |         | / W
      |         |/
      *---------*
          W

The box has a square base, W-by-W.
Its length (height) is L.

The girth is the distance around its "waist".
In this problem, the girth is \(\displaystyle 4W.\)

The postal restriction is: .\(\displaystyle L + 4W \,\le\,300\)

To maximize the volume, we use the entire 300 inches:
. . \(\displaystyle L + 4W \,=\,300 \quad\Rightarrow\quad L \:=\:300-4W\) .[1]

The volume is: .\(\displaystyle V \:=\:W^2L\)

Substitute [1]: .\(\displaystyle V \:=\:W^2(300-4W) \:=\:300W^2 - 4W^3\)

Set the derivative equal to zero: .\(\displaystyle 600W - 12W^2 \:=\:0\)

. . \(\displaystyle 12W(50-W) \:=\:0 \quad\Rightarrow\quad \rlap{//////}W = 0,\:W = 50\)

Substitute into [1]: .\(\displaystyle L \,=\,300-4(50) \,=\,100\)


Therefore, the package should be: .\(\displaystyle 50\!\times\!50\!\times\!100 \text{ inches.}\)

 

[Ghost3k]

thank you so much soraban! You are awesome. The girth part was the most confusing part. But your explanation was excellent, especially the drawing.