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• Level: GCSE
• Subject: Maths
• Word count: 3299

# The Open Box Problem

Extracts from this document...

Introduction

Math Coursework

The Open Box Problem

An open box is to be made from a sheet of card

Identical squares are to be cut off the four corners of the card as shown in the following diagram

1. For any sized square sheet of card, INVESTIGATE the size of the cut out square, which makes an open box of largest volume
2. For any sized rectangular sheet of card, INVESTIGATE the size of the cut out square, which makes an open box of largest volume

In this section of this investigation I am going to investigate using both trial and improvement and spreadsheets which size cutout will give the largest volume of the box.

Middle

27.6

4722.912

40

6.3

27.4

4729.788

40

6.4

27.2

4734.976

40

6.5

27

4738.5

40

6.6

26.8

4740.384

40

6.7

26.6

4740.652

40

6.8

26.4

4739.328

40

6.9

26.2

4736.436

40

7

26

4732

 S X B V 40 6.6 26.8 4740.38400 40 6.61 26.78 4740.48312 40 6.62 26.76 4740.56611 40 6.63 26.74 4740.63299 40 6.64 26.72 4740.68378 40 6.65 26.7 4740.71850 40 6.66 26.68 4740.73718 40 6.67 26.66 4740.73985 40 6.68 26.64 4740.72653 40 6.69 26.62 4740.69724 40 6.7 26.6 4740.65200
 S X B V 40 6.66 26.68 4740.73718 40 6.661 26.678 4740.73817 40 6.662 26.676 4740.73900 40 6.663 26.674 4740.73966 40 6.664 26.672 4740.74017 40 6.665 26.67 4740.74052 40 6.666 26.668 4740.74071 40 6.667 26.666 4740.74073 40 6.668 26.664 4740.74060 40 6.669 26.662 4740.74031 40 6.67 26.66 4740.73985

### Percentage Form Of X

 S X S2 X2 %age X 10 1.666 100 2.775556 2.775556 20 3.333 400 11.10889 2.777222 30 5 900 25 2.777778 40 6.666 1600 44.43556 2.777222

### Conclusion

I have found out using my tables that 2.77% of the total area is the best value for X2.

X2=2.77% of S2

To find the percentage value of all of the Xs you need to multiply 2.77 by 4. The 4 is all 4 corners that need to be cut out.

4X2 = (4 x 2.77)% of S2

4X2 = 11.08 % of S2

## The Rectangle

 S1 S2 X B1 B2 V 10 5 1 8 3 24 10 5 2 6 1 12 10 5 3 4 -1 -12 10 5 4 2 -3 -24 10 5 5 0 -5 0 10 5 6 -2 -7 84 10 5 7 -4 -9 252 10 5 8 -6 -11 528 10 5 9 -8 -13 936 10 5 10 -10 -15 1500
 S1 S2 X B1 B2 V 10 5 0 10 5 0 10 5 0.5 9 4 18 10 5 1 8 3 24 10 5 1.5 7 2 21 10 5 2 6 1 12 10 5 2.5 5 0 0 10 5 3 4 -1 -12 10 5 3.5 3 -2 -21 10 5 4 2 -3 -24 10 5 4.5 1 -4 -18
 S1 S2 X B1 B2 V 10 5 1 8 3 24 10 5 1.1 7.8 2.8 24.024 10 5 1.2 7.6 2.6 23.712 10 5 1.3 7.4 2.4 23.088 10 5 1.4 7.2 2.2 22.176 10 5 1.5 7 2 21 10 5 1.6 6.8 1.8 19.584 10 5 1.7 6.6 1.6 17.952 10 5 1.8 6.4 1.4 16.128 10 5 1.9 6.2 1.2 14.136 10 5 2 6 1 12
 S1 S2 X B1 B2 V 10 5 1 8 3 24 10 5 1.01 7.98 2.98 24.0182 10 5 1.02 7.96 2.96 24.03283 10 5 1.03 7.94 2.94 24.04391 10 5 1.04 7.92 2.92 24.05146 10 5 1.05 7.9 2.9 24.0555 10 5 1.06 7.88 2.88 24.05606 10 5 1.07 7.86 2.86 24.05317 10 5 1.08 7.84 2.84 24.04685 10 5 1.09 7.82 2.82 24.03712 10 5 1.1 7.8 2.8 24.024
 S1 S2 X B1 B2 V 10 5 1.05 7.9 2.9 24.0555 10 5 1.051 7.898 2.898 24.05571 10 5 1.052 7.896 2.896 24.05589 10 5 1.053 7.894 2.894 24.05603 10 5 1.054 7.892 2.892 24.05614 10 5 1.055 7.89 2.89 24.05622 10 5 1.056 7.888 2.888 24.05625 10 5 1.057 7.886 2.886 24.05626 10 5 1.058 7.884 2.884 24.05623 10 5 1.059 7.882 2.882 24.05616 10 5 1.06 7.88 2.88 24.05606

20 x 10

 S1 S2 X B1 B2 V 20 10 1 18 8 144 20 10 2 16 6 192 20 10 3 14 4 168 20 10 4 12 2 96 20 10 5 10 0 0 20 10 6 8 -2 -96 20 10 7 6 -4 -168 20 10 8 4 -6 -192 20 10 9 2 -8 -144 20 10 10 0 -10 0 20 10 11 -2 -12 264
 S1 S2 X B1 B2 V 20 10 2 16 6 192 20 10 2.1 15.8 5.8 192.444 20 10 2.2 15.6 5.6 192.192 20 10 2.3 15.4 5.4 191.268 20 10 2.4 15.2 5.2 189.696 20 10 2.5 15 5 187.5 20 10 2.6 14.8 4.8 184.704 20 10 2.7 14.6 4.6 181.332 20 10 2.8 14.4 4.4 177.408 20 10 2.9 14.2 4.2 172.956 20 10 3 14 4 168

Conclusion

n="1" rowspan="1">

5.66

192.3393

20

10

2.18

15.64

5.64

192.2969

20

10

2.19

15.62

5.62

192.2478

20

10

2.2

15.6

5.6

192.192

 S1 S2 X B1 B2 V 20 10 2.11 15.78 5.78 192.4497 20 10 2.111 15.778 5.778 192.4499 20 10 2.112 15.776 5.776 192.45 20 10 2.113 15.774 5.774 192.4501 20 10 2.114 15.772 5.772 192.4501 20 10 2.115 15.77 5.77 192.45 20 10 2.116 15.768 5.768 192.4498 20 10 2.117 15.766 5.766 192.4496 20 10 2.118 15.764 5.764 192.4493 20 10 2.119 15.762 5.762 192.4489 20 10 2.12 15.76 5.76 192.4485

30 x 15

 S1 S2 X B1 B2 V 30 15 1 28 13 364 30 15 2 26 11 572 30 15 3 24 9 648 30 15 4 22 7 616 30 15 5 20 5 500 30 15 6 18 3 324 30 15 7 16 1 112 30 15 8 14 -1 -112 30 15 9 12 -3 -324 30 15 10 10 -5 -500 30 15 11 8 -7 -616
 S1 S2 X B1 B2 V 30 15 3 24 9 648 30 15 3.1 23.8 8.8 649.264 30 15 3.2 23.6 8.6 649.472 30 15 3.3 23.4 8.4 648.648 30 15 3.4 23.2 8.2 646.816 30 15 3.5 23 8 644 30 15 3.6 22.8 7.8 640.224 30 15 3.7 22.6 7.6 635.512 30 15 3.8 22.4 7.4 629.888 30 15 3.9 22.2 7.2 623.376 30 15 4 22 7 616
 S1 S2 X B1 B2 V 30 15 3.1 23.8 8.8 649.264 30 15 3.11 23.78 8.78 649.3319 30 15 3.12 23.76 8.76 649.3893 30 15 3.13 23.74 8.74 649.4362 30 15 3.14 23.72 8.72 649.4726 30 15 3.15 23.7 8.7 649.4985 30 15 3.16 23.68 8.68 649.514 30 15 3.17 23.66 8.66 649.5191 30 15 3.18 23.64 8.64 649.5137 30 15 3.19 23.62 8.62 649.498 30 15 3.2 23.6 8.6 649.472
 S1 S2 X B1 B2 V 30 15 3.161 23.678 8.678 649.51496 30 15 3.162 23.676 8.676 649.51583 30 15 3.163 23.674 8.674 649.5166 30 15 3.164 23.672 8.672 649.51726 30 15 3.165 23.67 8.67 649.51782 30 15 3.166 23.668 8.668 649.51827 30 15 3.167 23.666 8.666 649.51862 30 15 3.168 23.664 8.664 649.51887 30 15 3.169 23.662 8.662 649.51901 30 15 3.17 23.66 8.66 649.51905 30 15 3.171 23.658 8.658 649.51899

40 x 20

 S1 S2 X B1 B2 V 40 20 1 38 18 684 40 20 2 36 16 1152 40 20 3 34 14 1428 40 20 4 32 12 1536 40 20 5 30 10 1500 40 20 6 28 8 1344 40 20 7 26 6 1092 40 20 8 24 4 768 40 20 9 22 2 396 40 20 10 20 0 0 40 20 11 18 -2 -396
 S1 S2 X B1 B2 V 40 20 4 32 12 1536 40 20 4.1 31.8 11.8 1538.484 40 20 4.2 31.6 11.6 1539.552 40 20 4.3 31.4 11.4 1539.228 40 20 4.4 31.2 11.2 1537.536 40 20 4.5 31 11 1534.5 40 20 4.6 30.8 10.8 1530.144 40 20 4.7 30.6 10.6 1524.492 40 20 4.8 30.4 10.4 1517.568 40 20 4.9 30.2 10.2 1509.396 40 20 5 30 10 1500
 S1 S2 X B1 B2 V 40 20 4.2 31.6 11.6 1539.552 40 20 4.21 31.58 11.58 1539.5818 40 20 4.22 31.56 11.56 1539.5978 40 20 4.23 31.54 11.54 1539.5999 40 20 4.24 31.52 11.52 1539.5881 40 20 4.25 31.5 11.5 1539.5625 40 20 4.26 31.48 11.48 1539.5231 40 20 4.27 31.46 11.46 1539.4699 40 20 4.28 31.44 11.44 1539.403 40 20 4.29 31.42 11.42 1539.3224 40 20 4.3 31.4 11.4 1539.228
 S1 S2 X B1 B2 V 40 20 4.22 31.56 11.56 1539.5978 40 20 4.221 31.558 11.558 1539.5986 40 20 4.222 31.556 11.556 1539.5993 40 20 4.223 31.554 11.554 1539.5999 40 20 4.224 31.552 11.552 1539.6003 40 20 4.225 31.55 11.55 1539.6006 40 20 4.226 31.548 11.548 1539.6007 40 20 4.227 31.546 11.546 1539.6007 40 20 4.228 31.544 11.544 1539.6006 40 20 4.229 31.542 11.542 1539.6003 40 20 4.23 31.54 11.54 1539.5999

### Percentage Form of X

 S1 S2 X S1 x S2 X2 %age of X2 10 5 1.057 50 1.117249 2.114 20 10 2.113 200 4.464769 1.0565 30 15 3.17 450 10.0489 0.7044444 40 20 4.226 800 17.85908 0.52825

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