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 Question from Dave, a teacher: We know that calculus can be used to maximise the volume of the tray created when cutting squares from 4-corners of a sheet of card and then folding up. What I want is to find the sizes of card that lead to integer solutions for the size of the cut-out, the paper size must also be integer. EG 14,32 cutout 3 maximises volume as does 13,48 cutout 3. I have done this in Excel but would like a general solution and one that does not involve multiples of the first occurrence, as 16, 10 cutout 2 is a multiple of 8,5 cutout 1.

Dave,

We have published a complete study (without calculus) if you start with a square sheet.
The cut out is 1/6 the dimensions of the square - so all multiples of 6 will produce the desired result.

Here is a Word file that describes a hands-on investigation of changes in volume, which also help foster key connections to important ideas of change and optimization.

Walter

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