The Open Box Problem
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The square cut outs will rise in measurements of 1cm at a time, beginning with a cut out of 1 cm, and 9cm being the maximum for there to be a box left.
I will then investigate various other ratios.
I have again constructed a spreadsheet to assist with my workings out. Firstly I am looking at whole numbers, a beginning with a width of 20cm and a length of 40cm, and cut outs from 1 to 9cm.
Here are the results I got.
You should be able to see from this table, and the graph I have drawn that that the largest volume achieved is 1536cm², and this is obtained when the amount cut off from the corners measures 4cm². I can also see that to make my results more accurate, to more decimal places, I next need to look between 4 and 5cm² cut offs.
Below are the formulae I used for the above spreadsheet, which only differs slightly to the one that was used for the square. All that has changed is the length being 40, which is reflected below.
Next are my results from calculating the volume when the cut out area is measured to one decimal place.
I have also drawn a graph to illustrate my results.
From these results, where the largest volume of 1539.522cm² is achieved with a cut o ...
