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In my optimization class, we are now covering the basics which are just simple word problems to find absolute minima and maxima using calculus, but the wording of this problem has me completely confused:

We have a piece of cardboard that is $50$ cm by $20$ cm and we are going to cut out the corners and fold up the sides to form a box. We are asked to determine the height of the box that will give a maximum volume.

Despite my best efforts, I cannot figure out how the box is to be constructed from the piece of paper (cutting the corners). Can someone please show me how to get the function I am supposed to maximize and how to get there? I thank all helpers.

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    $\begingroup$ When the box's corners are cut, a square is cut out from it. This gets you 2 new edges per corner which u can fold up to meet to make the box. $\endgroup$ Sep 15 '20 at 4:38
  • $\begingroup$ @SharkyKesa thank you kindly $\endgroup$
    – kroner
    Sep 15 '20 at 4:47
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Refer to the picture:

enter image description here

Notice the $4$ corners, let those be of length $h$ (height), after cutting them up, we fold the $4$ sides up. Resulting in a base area of $(a-2h)(b-2h)$, you should be able to obtain the formula of volume from there and optimize it.

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  • $\begingroup$ thank you very much so is the formula for volume $(a-2h)(b-2h)h$? $\endgroup$
    – kroner
    Sep 15 '20 at 4:46
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    $\begingroup$ yes, that is correct. $\endgroup$ Sep 15 '20 at 4:58
  • $\begingroup$ thank you very much $\endgroup$
    – kroner
    Sep 15 '20 at 5:02

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