The Open Box Problem
Introduction
The aim of my algebraic investigation into the open box problem is to
determine the size of square cut which makes the volume of the box as
large as possible for any given rectangular sheet of card. The problem
itself is simple, an open box is made from a sheet of card, identical
squares are then cut off each of the four corners, the sheet is then
folded to make box. It is my aim to find out the maximum square cut
which gives me the maximum volume box.
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Strategy
1. Try to find the size of cut-out that will give me the maximum
volume of a piece of card 6cm x 6cm, progressing onto algebra.
2. Look for limitations in the results.
3. Devise a plan for overcoming these limitations.
4. Extend the work to find a general formula that will help me to
work with all squares and rectangles.
Technique
In this situation there is no need for us to go even close to a pair
of scissors or piece of card, for this investigation I am to use
Excel. Excel is a computer programme in which I can input information;
it will then calculate this information and give me results for what
different size cut outs for different sized card.
The set-up of this is quite simple;
Size of card is out into cell c3 in this case
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Height equals the size of the cut-out so corresponding cell number is
typed in.
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Volume is calculated
With a real box by
Size of cut-out in Width x Depth x Height
Cells a6, a7 etc… The same happens here
Corresponding cell
Numbers put in with * between each one meaning multiply.
Width is calculated using formula.
$ is used to keep the term c3