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The Open Box Problem

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Introduction

Part 1: Square Ryan Simmons For any sized square sheet of card investigate the size of the cut out square that makes an open box of the largest volume. The first square to be tested has measurements of 30 x 30 cm. For cut sizes I will start from the smallest (whole) number possible (1cm) I will then work my way up to find which size cut gives the box the largest volume. So far as the size of the cuts increase the volume increases. I predict that a cut size of 3 x 3 cm will give an even bigger volume for the box. A cut of 3 x 3 cm gives a volume of 1728 cm3, hence my prediction was right. 3cm x 24cm x 24cm = 1728 cm� The prediction was right and so far there is no obvious pattern between the cut sizes and the volume of the box. To save me from drawing a diagram for every cut size I will record my results in a table Cut size (cm) W L V (cm3) 4 22 22 1936 5 20 20 2000 6 18 18 1944 7 16 16 1792 8 14 14 1568 This is a spreadsheet, where the value of the volume is a product of the cut size, the width and the length. ...read more.

Middle

10cm is 1 / 6 of 60cm. Cut size (cm3) W L V (cm 3) 7 46 46 14812 8 44 44 15488 9 42 42 15876 10 40 40 16000 11 38 38 15884 12 36 36 15552 13 34 34 15082 This table shows the volumes of different cut sizes for a piece of card with dimensions of 60 x 60 cm. The prediction was right. However, this could mean that only numbers divisible by 6 have this rule applied to them. To see if the prediction was right for any piece of card I will try a box with lengths of 50 x 50 cm. As this number is not divisible by 6 my prediction may prove incorrect. Again I will record my results in a spreadsheet. Cut size (cm3) W L V (cm 3) 5 40 40 8000 6 38 38 8664 7 36 36 9072 8 34 34 9248 8.3 33.4 33.4 9259.148 9 32 32 9216 From the table it can be seen that a cut size of 8.3cm (1/6 of 50) gives the largest volume. Therefore my prediction was correct: A cut size that is 1/6 of the length produces the greatest volume. I will not investigate cut sizes greater than 9cm as the volume begins to decrease. ...read more.

Conclusion

The volume was 192.4497cm3 . From the spreadsheet I produced a graph to show the relationship between cut size and volume. The blue line on the graph shows that the maximum volume is 192.4497cm3 when the cut size is 2.11cm. I then carried out the same procedure for a rectangle that is 30cm x 10cm. Cut size Length Width Volume 0 30 10 0 1 28 8 224 2 26 6 312 2.1 25.8 5.8 314.244 2.2 25.6 5.6 315.392 2.25 25.5 5.5 315.5625 2.26 25.48 5.48 315.5647 2.27 25.46 5.46 315.5563 3 24 4 288 4 22 2 176 5 20 0 0 The spreadsheet shows that the maximum volume is 315.5647cm3 when the cut size is 2.26cm. From the spreadsheet I produced a graph to show the relationship between cut size and volume. The blue line on the graph shows that the maximum volume is 315.5647cm3 when the cut size is 2.26cm. Conclusions so far: For a rectangle that is 20cm x 10cm the cut size that produces the greatest volume is 2.11cm. For a rectangle that is 30cm x 10cm the cut size that produces the greatest volume is 2.26cm At present I can see no relationship between the cut size and the dimensions of the piece of rectangular card. This is possibly because, unlike the square in Part 1, the rectangle has 2 sides of different dimensions. ?? ?? ?? ?? 1 ...read more.

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