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

• Nov 1st 2012, 03:48 AM
au79
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?

Attachment 25502
• Nov 1st 2012, 02:35 PM
skeeter
Re: Find the maximum volume, from an a4 sheet of paper folded into a box
V = x(210-2x)(297-2x)

volume in mm3
• Nov 1st 2012, 03:45 PM
au79
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)
• Nov 1st 2012, 03:52 PM
MarkFL
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.
• Nov 1st 2012, 10:41 PM
au79
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 =
• Nov 1st 2012, 10:56 PM
MarkFL
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:

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