To find out what size squares must be cut out of the corners of square and rectangle pieces of card to give the box it would create its optimum volume.

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As a class we were given a mathematical problem to solve as our first bit of maths coursework for GCSE, it is called the open box problem. We have been asked to find out what size square we would have to cut out the corner of a piece of card, either square or rectangular, to find the optimum volume of that box. I will start at 1cm³ Square and work up in 1cm³. 

         I would expect the optimum volume to most of the time be in the decimal places so I will have to look at all the decimal places between the highest volume I have and the highest one on one of its sides. During this experiment I will be looking for relationships between the length of the side and the volume of the box.

Aim:-  To find out what size squares must be cut out of the corners of square and rectangle pieces of card to give the box it would create its optimum volume.

Prediction:- I predict that the formulas that I find for the squares will be completely different to the formulas for the rectangles. I predict that after we have found the optimum volume that the volumes after that will drop very rapidly.  I also think that the size square that we have to cut out of the corner to create the box will always be quite a low number around two or three because as you cut bigger squares off the corner you lose a lot more card that could be used to make the box bigger, but when you cut a little off the corners you get much depth so the volume is compromised. But when you cut around two or three cm³ you will have just about the right amount of height and length to give the boxes optimum volume.

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        An open box is to be made from a sheet of card. Identical squares are cut off the four corners of the card as shown in the diagram below.

           10cm

 

10cm

The card is then folded along the dotted lines to make the box. The main aim of this activity is to determine the size of the square cut out which makes the volume of the box as large as possible for any give square of rectangular piece of card.

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