Maths Coursework - The Open Box Problem

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Maths Coursework – The Open Box Problem

In this investigation we are asked to determine the size of the square cut out which makes the volume of the box as large as possible for any given rectangular sheet of card. This investigation is dived up into 2 parts; Part 1 and Part 2.

Part 1

For this part we are asked to determine for any sized square sheet of card, investigate the size of the cut out square which makes an open box of the largest volume.

Part 2

For this part we are asked to determine for any sized rectangular sheet of card, investigate the size of the cut out square which makes an open box of the largest volume.

Part 1

I decided to construct a table to investigate what length of cut out would give the largest volume.

This is the table I constructed for a 10 by 10 square using the program Microsoft Excel.

20 by 20 square

 

I have now found out that in a 10 by 10 square the cut out which gives the largest volume is between 3 and 4 but more towards 3 since 3 gives a larger volume. To obtain a more accurate result I will zoom in the shaded region.

I will zoom in again to increase the degree of accuracy in finding out what length of cut out gives the largest volume.

I have decided to stop zooming in since the length of the cut out, is in 3 decimal place, which is a reasonable degree of accuracy.

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This is a graph I produced of the program Autograph. I have made it bold so that it is easier to interpret and I have also added the maximum value in a pink text box and the equation of the graph in a blue text box.

From the graph I can see that for a 10 by 10 square the cut out, to 3 decimal place, which gives the maximum volume is 3.333.

30 by 30 square

I have now found out that in a 10 by 10 square the cut out which gives the largest volume is ...

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