Tbe Open Box Problem

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The open box problem        By Luke Johnston

 An open box is to be made from a sheet of card with identical squares cut of the corners to make a box:

                       y

x = corner length

y = box length

The card is then folded along the dashed lines to make the box. The aim of this activity is to determine the size of the cut out square which makes the largest volume for a square.

Firstly I will find this in a square of y = 10cm

To find the volume I will use this equation-Volume= Width x length x height

 

The maximum box volume is made from the cut out square of 2cms. I will now try between 1-2 cm because the highest volume is somewhere between them.

The highest value of x is 1.7, but once again I will find a volume even higher in the 10 by 10 box by going into two decimal places.

The highest value of x is therefore 1.67. I will now go into even smaller numbers between 1.66 and 1.67.

The highest is 1.667. I will show the graph to show the change in volume for a 10 by 10 box.

The maximum value of x to make the highest volume is 1/6 of the length. I can tell this because 10 divided by 6 is 1.666666667. I will now try a 15 by 15 square. I shall predict the maximum volume of the square by dividing 15 by 6 and this makes 2.5.

I will find the maximum value of x between 2 and 3 as it is between them.

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I shall now try between 2.4 and 2.6.

The maximum value of x is 2.5cm. This shows my prediction was correct. This is that a square box has to be divided by six to find the cut out size in which would give the maximum volume.

To help find the highest value of x to make the largest volume, I will use algebra, with aid from textbook sources. I will need to divide the cut out square by the original length of the box.

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