• 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.

but were separated so as not to read as a full text. Trial one consisted of paragraphs form one article and the paragraphs were presented only after the single words and sentences had already been read. In trial two which consisted of paragraphs taken from a different article the paragraphs

2. ## Leaves Project

The box plots show us that the two "whiskers" are almost equal in all of the box plots; this shows us that they are of an even distribution. This supports my forth hypothesis: The length and width of the leaves will follow a normal to almost normal distribution.

1. ## Data Handling Project

Alexander Solzhenitsyn. Lord Byron. They should be counted as some newspapers may then make abbreviations for them, as they still have been included in the article. Newspaper Headlines, should they be counted. Pout is Out (The Sun) Headlines will not be counted as they are an attempt to grab attention, I will focus more on the article.

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

upper shore, chosen depending on the outcome of the results of my preliminary investigation. I will need to make the starting spot random. So wherever I am standing I will generate a random number using my calculator and if it is even my starting position will be to my right,

1. ## 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

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

13 93 44 14 93 44 15 95 45 16 96 45 17 97 46 18 100 47 Median 19 103 47 20 105 50 21 110 50 22 110 50 23 110 50 24 111 55 25 112 57 26 115 58 27 119 58 Upper Quartile 28 120

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. ## 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 )

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