• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
• 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.

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.

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

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

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

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

Conclusion

3 /27

=9259.259cm3

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

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

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

This student written piece of work is one of many that can be found in our GCSE Comparing length of words in newspapers section.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related GCSE Comparing length of words in newspapers essays

1. ## Assesment of Reading Difficulties in Patient AM Following the Development of Vascular Dementia.

AM to that of age matched controls are both clear indicators that AM is an attentional dyslexic further supported by his reported inability to follow the content of a text even when read at a sentence level, despite his maintained ability to read both sentences and paragraphs accurately albeit slowly.

2. ## The Open Box Problem

x 20 x 5 = 2,000cm� 6cm 18cm 18cm 18 x 18 x 6 = 1,944cm� 7cm 16cm 16cm 16 x 16 x 7 = 1,792cm� 8cm 14cm 14cm 14 x 14 x 8 = 1,568cm� 9cm 12cm 12cm 12 x 12 x 9 = 1,296cm� 10cm 10cm 10cm 10

1. ## Open Box Problem.

44.3556 74.07385 1.68 6.64 44.0896 74.07053 1.69 6.62 43.8244 74.06324 1.7 6.6 43.56 74.052 The maximum volume I found in this table was at 1.67cm with its volume at 74.07385cm Size of square cut = 20 by 20 (cm) Length cut from each side ( x )

2. ## Investigate if there is a relationship between the length and width of the leaves.

The scatter graph indicates a strong positive correlation, as the points are close to a straight line. As the length increase the width also increases. I shall work out the Product Moment Correlation Coefficient (PMCC) As a check I will use the function key on the computer to find the PMCC.

1. ## Swimming Problem Maths Investigation.

25 95 165 235 305 375 445 515 585 655 2 30 100 170 240 310 380 450 520 590 660 3 35 105 175 245 315 385 455 525 595 665 4 40 110 180 250 320 390 460 530 600 670 5 45 115 185 255 325 395

2. ## Data Handling Project

Section Pages News 7 Entertainment 3 Business 6 Sport 2 Other (Advertisements) Total 0 18 I then take away the number of Other (Advertisements) pages there are. Which in this newspaper is none, so nothing needs to be done. Then to work out how many words should be collected in the News section, you simply, do the formula: No.

1. ## This investigation looked to see whether the height on the shore would affect the ...

I will need a 1/4 m2 quadrat. This will be moved along my continuous horizontal transect as a boundary for my selection area for topshells. I think 1/4 m2 is a suitable area for selection. It were smaller I would have to move a large distance to collect enough topshells to have good results.

2. ## The Open Box Problem

1128494 40.391 216.218 129.218 40.391 1128495 40.392 216.216 129.216 40.392 1128495 40.393 216.214 129.214 40.393 1128495 40.394 216.212 129.212 40.394 1128495 40.395 216.21 129.21 40.395 1128495 40.396 216.208 129.208 40.396 1128495 40.397 216.206 129.206 40.397 1128495 40.398 216.204 129.204 40.398 1128495 40.399 216.202 129.202 40.399 1128495 40.401 216.198 129.198 40.401

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to