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Find the maximum volume, from an a4 sheet of paper folded into a box

Grab a piece of paper. Find the maximum volume obtainable, by cutting corners out of the box.

I haven't done this math before, and It was part of a puzzle based question for a topic I'm doing.

and the funny thing is this is a particular component of math I'm doing next year.

We know that all cuts must be identical to make an open cut box.

Assuming dimensions of 297mm and 210mm.

The maximum allowable cut is 210mm/2 = 105mm (although anything near would be practically useless)

Basically I have so far. where do I go from here?

I've plugged the values in and I get 24x-2028 = 2028/24 = 84.5mm.

I plugged this into wolfram, but I did get the 2028 somewhere in my working out. oh yeah it was -2x * by length and by width. am I on the right track?

I'm not right the answer is around about 40mm according to my spreadsheet.

Attachment 25502

Re: Find the maximum volume, from an a4 sheet of paper folded into a box

V = x(210-2x)(297-2x)

volume in mm^{3}

Re: Find the maximum volume, from an a4 sheet of paper folded into a box

Sorry, I do mean maximum volume, but I need to determine the length of cuts (or the height)

Re: Find the maximum volume, from an a4 sheet of paper folded into a box

To find the actual value for x which yields the maximal volume will require differential calculus. Otherwise, graph the volume function (as given by **skeeter**) and estimate the value of x that is at the maximum value for the volume.

Re: Find the maximum volume, from an a4 sheet of paper folded into a box

Thanks....I've done that in excel, graphed it and estimated the max point to be 40mm.

What do I need to do to get an accurate answer using calculus. I've got my formula in the first post but I don't know where to go from there.

Alternatively I came up with another solution, I got from an example on Youtube.

And I got v(x) = 62370 -2028x + 12x^2

But I'm not sure how to plug this in to get my answer. The example on youtube was a nice neat problem with the width and length =

Re: Find the maximum volume, from an a4 sheet of paper folded into a box

To find the exact answer using calculus, we would equate the derivative of the volume function to zero:

$\displaystyle V'(x)=6(2x^2-338x+10395)=0$

Take the appropriate root:

$\displaystyle x=\frac{169-\sqrt{7771}}{2}\approx40.4233621971911$