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  • Level: GCSE
  • Subject: Maths
  • Word count: 1188

The open box problem

Extracts from this document...

Introduction

The open box problem

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 below.

image00.png

The card is then folded along the dotted lines to make the box.  

The aim of this project is to determine the size of the square cut out in any given size rectangle sheet of card with the largest volume.

Task 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.

For this task I will use 20cm*20cm square sheet of card. The formula used to get a volume is V=length x Width x height

...read more.

Middle

1.9

498.636

20

20

2

512

20

20

2.1

524.244

20

20

2.2

535.392

20

20

2.3

545.468

20

20

2.4

554.496

20

20

2.5

562.5

20

20

2.6

569.504

20

20

2.7

575.532

20

20

2.8

580.608

20

20

2.9

584.756

20

20

3

588

20

20

3.1

590.364

20

20

3.2

591.872

20

20

3.3

592.548

20

20

3.4

592.416

20

20

3.5

591.5

image02.png

now for more accuracy I will use 3 decimal place and I will concentrating in between 3.332 and 3.334 . this time the results should be more accurate then any other charts.

length

width

height

volume

20

20

3.332

592.5925

20

20

3.333

592.5926

20

20

3.334

592.5926

20

20

3.335

592.5925

20

20

3.336

592.5923

20

20

3.337

592.5921

20

20

3.338

592.5917

20

20

3.339

592.5913

20

20

3.340

592.5908

20

20

3.341

592.5902

image03.png

from the graph the size 3.333 appears to be the highest which means it is the one size that gives out the largest volume.

To find other size to be cut out on other size of paper, I divide 3.333 by 20 making it 0.16665 and in fraction 1/6.

And that answer helped me to make a formula as 1/6 of the 20 cm makes the highest volume so I made this formula:

Volume= V

Length= L

Width= W

Height= H

As length and width are the same in a square we are only going to use length

  • V=  (L-2L/6)(L-2L/6) x L/6

Simplified:

  • V= (L2 - 2L2/6 - 2L2/6 + 4L/36) x L/6
  • V= L3/6 -2L3/36 – 2L3 /36 + 4L2/216

Now I have put the denominator as 216

  • V= (4L3 + 36L3 - 12L3 - 12L3  )/216
  • V= 40L3 – 24L3 /216
  • V= 16L3 /216

Simplified:

  • V= 2L3/27
...read more.

Conclusion

3 /27

=9259.259cm3

Task2

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

For this task I will be investigating in rectangular sheet of paper. I will be using 20cm of width and 40cm of length in the ratio of 1:2. as from the square investigation I will be cutting out sizes from 1-10cm on each side.

Length

Width

Height

Volume

40

20

1

684

40

20

2

1152

40

20

3

1428

40

20

4

1536

40

20

5

1500

40

20

6

1344

40

20

7

1092

40

20

8

768

40

20

9

396

40

20

10

0

image04.png

In this chart the highest volume is 1536cm3 by cutting out 4 cm on each side. It is possible to get more accurate results it will be calculated to 1 decimal place. I will be concentrating between 3 to 5for best results.

Length

Width

Height

Volume

40

20

3

1428

40

20

3.1

1445.964

40

20

3.2

1462.272

40

20

3.3

1476.948

40

20

3.4

1490.016

40

20

3.5

1501.5

40

20

3.6

1511.424

40

20

3.7

1519.812

40

20

3.8

1526.688

40

20

3.9

1532.076

40

20

4

1536

40

20

4.1

1538.484

40

20

4.2

1539.552

40

20

4.3

1539.228

40

20

4.4

1537.536

40

20

4.5

1534.5

40

20

4.6

1530.144

40

20

4.7

1524.492

40

20

4.8

1517.568

40

20

4.9

1509.396

40

20

5

1500

image05.png

 in this graph and chart data, the highest point is 1539.552cm3 from 4.2 cm cut out. Now I will be calculating the results in three decimal places

...read more.

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