# The Open Box Problem

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Introduction

Mathematics GCSE The Open Box Problem Tiers F, I and H Introduction 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 figure 1. Figure 1: 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 given rectangular sheet of card. 1. For any sized square sheet of card, investigate the size of the cut out square which makes an open box of the largest volume. 2. For any sized rectangular sheet of card, investigate the size of the cut out square which makes an open box of the largest volume. Question 1 I began work on question 1, which was to investigate the correct cut out size to satisfy the largest possible volume of the open box on any sized square piece of card. I started by using a square piece of card measuring 10 x 10. To work out the volume for the open box with the variable being 'x' I made a formula to help: V=x(l-2x)(l-2x) Here are my results: size of cut out 'x' (cm) volume (cm�) 1 64 2 72 3 48 4 16 The highest volume is 72cm� with a cut out size of 2cm. ...read more.

Middle

To work out the maximum volume for a square piece of card substituting 'x' with 1/6L: V=(1/6L)L�-4(1/6L)�L+4(1/6L)� V=1/6L�-4/36L�+4/216L� V=1/6L�-1/9L�+1/54L� V=9/54L�-6/54L�+1/54L� V=4/54L� V=2/27L� Maximum volume=2/27L� This formula can be used to work out the maximum volume of any open box from a square shaped piece of card without needing to work out anything to do with the cut out size. Question 2 I them moved on to question 2, which was to investigate the correct cut out size to satisfy the largest possible volume of the open box on any sized rectangular piece of card. I had already found the cut out size for a rectangle of ratio 1:1 in the first question, as a square is also a type of rectangle. I first tried the formula for ratio 1:2 rectangles: To work out the volume of the open box from a rectangular piece of card the formula is: V=(KW-2X)(W-2X)xX I investigated the ideal cut out size for a rectangle 16 x 8: size of cut out 'x' (cm) volume (cm�) 1 84 1.5 97.5 2 96 2.5 82.5 3 60 3.5 31.5 4 0 I then homed in between 1.5 and 2cm for 'X': size of cut out 'x' (cm) volume (cm�) 1.5 97.5 1.6 98.304 1.7 98.532 1.8 98.208 1.9 97.356 2 96 I then homed in once more with values of 'X' to two decimal places: size of cut out 'x' (cm) ...read more.

Conclusion

volume (cm�) 1.5 157.5 1.6 159.744 1.7 161.092 1.8 161.568 1.9 161.196 2 160 I then homed in again between 1.8 and 1.9cm for 'x': size of cut out 'x' (cm) volume (cm�) 1.8 161.568 1.81 161.568564 1.82 161.560672 1.83 161.544348 1.84 161.519616 1.85 161.4865 1.86 161.445024 1.87 161.395212 1.88 161.337088 1.89 161.270676 1.9 161.196 The highest volume for this rectangular sheet of card is 161.568564cm� when 'X' is to two decimal places being 1.81cm. I then divided 8 by 1.81 and got the answer 4.419889503. This is very similar to the answer for rectangular sheet 18x6 so this is proof that for rectangles with ratio 1:3 the cut out size is about 1/4 when rounded down from 4.4.... I now have results for rectangles with 3 different ratios and I have created this table: ratio cut out size if width=1 fraction 1:1 0.16666 1/6 1:2 0.21111 about 1/5 1:3 0.22666 about 1/4 I investigated some formulae to find the cut out size easily with less calculation. This rectangular problem can be solved algebraically: Area=(KW-2X)(W-2X) Volume=(KW-2X)(W-2X)xX =(KW�-2KWX-2XW-4X�)xX =KW�X-2KWX�-2X�W-4X� Gradient=KW�-4KWX-4XW+12X� 0=12X�-4WX(K+L)+KW� A=12 B=-4W(K+1) C=KW� General Formula: X=-B�� B�-4AC 2A X=4W(K+1) � -(4W(K+1))�-48KW� 24 X=4W(K+1) � 16W�(K+1)(K+1)-48KW� 24 X=4W(K+1) � 16W�(K�+2K+1)-48KW� 24 X=4W(K+1) � 16K�W�+32KW�+16W�-48KW� 24 X=4W(K+1) � 16K�W�-16KW�+16W� 24 X=4W(K+1) � 16W�(K�-K+1) 24 X=W(K+1) � 4W K�-K+1 24 X=W(K+1) � W K�-K+1 6 X=cut out size W=Width K=Integer that is multiplied by 'W' to get the length. This formula can be used to calculate the maximum cut out section of any size rectangular sheet of card, in order to obtain the maximum volume. ...read more.

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