Application of the Derivative

[jacque1228]

The U.S. POstal Service stipulates that any boxes snet through the mail must have a length plus girth totaling no more than 108 in. Find the dimensions of th box with maximum volume that can be sent through the U.S. mail assuming that the width and the height of the box are equal.

 

[soroban]

Hello, jacque1228!

The U.S. POstal Service stipulates that any boxes sent through the mail
must have a length plus girth totaling no more than 108 in.
Find the dimensions of the box with maximum volume that can be sent through the U.S. mail
assuming that the width and the height of the box are equal.

          * - - - - - *
        /           / |
      /           /   |x         The length of the box is L
    * - - - - - *     |
    |           |     |               The girth is 4x.
   x|           |     *
    |           |   /x
    |           | /
    * - - - - - *
          L

We are told that: \(\displaystyle \;L\,+\,4x\:\leq\:108\)
. . To get maximum volume, we'll use the entire 108.
. . We have: \(\displaystyle \;L\,+\,4x\:=\:108\;\;\Rightarrow\;\;L\:=\:108\,-\,4x\;\) [1]

The Volume is Length x Width x Height, so we have: \(\displaystyle \;V\;=\;x^2L\)

Substitute [1]: \(\displaystyle \;V\:=\:x^2(108\,-\,x^2)\)

. . and that is the function we must maximize . . .

 

[Gene]

Volume = Length*Width*Height
W=H
girth = 2(W+H)
2(W+H)+L=108
4H+L=108
H=(108-L)/4
V=LWH = H²L = ((108-L)/4)²*L
Can you find dV/dL from there

 

[stapel]

What is the volume formula for a rectangular prism?

You have width w equals height h, so w = h. You also have length L.

"Girth" being the perimeter of the cross-section, you have:

. . . . .108 = 2w + 2h + L

Solve for "L=". Substitute using the provided equality condition, and plug into the "volume" formula. Then maximize.

If you get stuck, please reply showing how far you have gotten. Thank you.

Eliz.

 

[Gene]

Soroban's derivitive is friendlier than mine but there is a typo. It should, of course, be
V = x²(108-4x)

 

[TchrWill]

The rectangular area enclosing the maximum area for a given perimeter is a square.

Similarly, the rectangular solid enclosing the maximum volume for a given girth is a cube.

Therefore 108/3 = 36 = the dimensions of each side.

 

[Gene]

Will has an interesting bit of trivia but it doesn't apply to this problem. 36*4 = 144 > 108 so it can't be a cube.
-----------------
Gene