Addmath 2012 (Derek's)

7/31/2019 Addmath 2012 (Derek's)

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Sekolah Menengah KebangsaanKolombong, Kota Kinabalu, Sabah.

Additional MathematicsProject Work 2/2012

Name : Derek Duane OsmondClass : 5 PerdanaI/C Number : 950112-12-5369Teacher : Shirney Chua


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No. Topic Page

1 Acknowledgement 2

2 Objectives 3

3 Introduction 4-5

4 Part 1 6-8

5 Part 2 8-12

6 Part 3 13-17

7 Further Exploration 17-18

8 Reflection 19


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ACKNOWLEDGEMENTS

First of all, I would like to express my special thanks to Madam Shirney Chua, myAdditional Mathematics teacher, who gave me the opportunity to do this project and helpedme a lot throughout finishing this project. Without her guide, I may not finish my project anddo it properly.

Secondly, I would like to thank my parents and my family for providing everything,such as money to buy anything that are related to this project and their advises, which is themost needed to do this project. I am grateful for their constant support and help. Not forgottento my friends who have contributed lots of ideas in finding the topic that would be interestingto do and give their comments on my research. I really appreciate their kindness and help.

Besides that, I want to thank the respondents for helping and spending their time toanswer my questions for this project. Without the respondents, I might not be able tocomplete this project and because of their co-operation in answering the questions, I have theconclusion for this project.

Last but not least, I would like to express my gratitude to those who are involvedeither directly or indirectly in completing this project. Thank you for all the co-operationgiven.


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We students who are taking Additional Mathematics subject are required to carry out aproject work while we are in form 5 . This Year the Co-curiculum Development Division,Ministry of Education has prepared a tasks for us. We complete this task based on our area of interest. This project can be done in group or individually, and gladly to do this individually.Upon completion of the Additional Mathematic Project Work, we are to gain valuableexperiences and able to:

Apply and adapt a variety of problem solving strategies to solve routine and non-routine problems.

Experience classroom environments which are challenging, interesting andmeaningful and hence improve their thinking skills.

Experience classroom environment where knowledge and skills are applied inmeaningful ways in solving real-life problems.

Experience classroom environments where expressing ones mathematical thinking,reasoning and communication are highly encouraged and expected.

Experience classroom environments that stimulate and enhance effective learning.

Acquire effective mathematical communication through oral and writing, and to usethe language of mathematic to express mathematical ideas correctly and precisely.

Enhance acquisition of mathematical knowledge and skills through problem-solvingways that increase interest and confidence.

Prepare ourselves for the demand of our future undertaking and in workplace.

Realise that mathematics is an important and powerful tool in solving real-life

problems and hence develop positive attitude towards mathematics.

Train ourselves not only to be independent learners but also to collaborate, tocooperate and to share knowledge in an engaging and healthy environment.

Use technology especially the ICT appropriately and effectively.

Train ourselves to appreciate the intrinsic values of mathematic and to become morecreative and innovative.

Realize the importance and the beauty of mathematics.


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The history of packaging drinks dates back as early as the 17 th century. Back then, Waters

from natural springs were recognized as being safe (even healthy) to drink from earliest times

and were transported by wherever means that were available. Naturally carbonated waters

were collected into earthenware containers which were tightly sealed with cork and wax,

usually not very successfully. The used of earthenware bottles proved to be unsatisfactory for

the more highly carbonated aerated mineral waters and they were soon replaced by glass

bottles.

Many of the early glass bottles had round bottoms ensuring that they were stored on their

side, thereby keeping the corks moist and so preventing leakage from corks drying out. The

manufacture of glass bottles was a skilled job as they were hand blown. Although some semi-

automation had been introduced earlier, the first patent for an automatic glass bottle blowing

machine was granted to Michael J. Owens in the USA in 1904.High pressure generated inside

bottles by the carbonation caused frequent leakage and although improved by wiring-in-place, corks were generally unsatisfactorily. Many alternative forms of seals were patented

over the years and these fell broadly into three main categories:

Wire and rubber sealing devices were especially popular in the USA until the early 1900s.

The wire could be either an internal spring form, which held a seal in place on the inside of

the neck, or of the external swing type, in which an external wire frame was used to hold aceramic plug in place against a rubber seal. First patented by Charles de Quillfeldt in 1874,

this latter type is still currently in use for some specialty.

Variations on the theme of using an internal ball made from rubber, ebonite or glass were

developed and used with varying degrees of success. The ball was held in place by the

internal pressure. The most successful of these was patented by Hiram Codd of London. His

bottle was widely used in the UK from 1870s until the 1930s. A similar bottle, but with a

floating rubber ball acting as seal, was patented in the USA by S. Twitchell in 1883.


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The third popular alternative was the internal screw top bottle. Unlike todays bottle, thethread was on the inside of the bottle neck and an ebonite or wooden stopper screwed on to

the neck, with a rubber washer being used to improve the seal. These types of stoppers were

in common usage well into the 1950s in the UK. Ebonite, an early type of plastic resin

material soon replaced wood, which has a tendency to absorb moisture, causing it to swell

and crack the bottle neck.

A major step forward in sealing development was made by William Painter, who in 1895

patented the Crown Cork, founding the Crown Cork and Seal Company in 1 April 1892.Although initially slow to gain acceptance for two reasons:

The existing large capital investment in returnable bottles and bottling plant, and

The need for a tool to remove the crown, the crown cork eventually became popular,

especially for small single serve and beer bottles. Screw stopper retained their popularity for

the larger bottles where re-sealability was important.

Except for specialty earthenware ginger beer containers, glass bottles were the only form of

packaging for carbonates for over hundred years until the introduction of cans in the 1960s.

Then, just as the second half of the nineteenth century had been the time for product

development, the second half of the twentieth century became the time for packaging and

distribution of development.


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Packaged drinks are commonly served during parties, celebrations or events. There are many types of

packaged drinks sold in our country.

a) List the types of packaged drinks presently available in our country and their manufacturers.

b) Collect pictures of at least five different packaged drinks and their prices that you can find in the

supermarkets or shops. Your answer should include at least one type of Sabah products.

Types of packaged drinksBRANDS MANUFACTURERNestle - MILO Nestle Products Sdn. Bhd.Ribena Blackcurrant Malaysia Milk Sdn. Bhd.MARIGOLD - Orange Malaysia Milk Sdn. Bhd.F&N - Chrysanthemum Tea F&N Beverages Manufacturing Sdn. Bhd.

Desa Milk Desa Cattle(S) Sdn. Bhd.Dutch Lady - Strawberry Milk Industries Sdn. Bhd.Drinho - Ice Lemon Tea Ace Canning Corporation Sdn. Bhd.Yeos Justea Lemon Green Tea Yeo HianSeng(M) Sdn. Bhd.Dairy Milk Sabah Milk Industries Sdn. Bhd.Pokka Mango Milk Pokka Corporation(S) Limited Malaysia Branch

Brand Picture Price (RM)

Nestle - MILO(200ml)

1.50


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Yeos LemonGreen Tea(250ml)

2.00

MARIGOLD-Orange (250m)

1.50

F&N SeasonsChrysanthemumTea (250ml)

1.50


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Maju Jaya Sdn. Bhd. is a newly formed company that manufactures packaged drinks. As an assistantmarketing manager, you are required to carry out a market survey.

(a) Based on your survey, list five types of packaged drinks that are most popular among the students.Collect data on the preferred packaged drinks of at least 50 students, and complete Table 1.

No.

Table 1

No. Types of drinksStudentss name

PokkaMango Milk

Dutch LadyStrawberry

Yeos LemonGreen Tea

MarigoldOrange

F&NChrysanthemum

1 Mohd. Hafiz Hassim 2 Tommy Suin 3 Wesley Jusih 4 Boniface Allen V. 5 Muhd. Fikri 6 Aubrey Wong 7 Amelda Evylisa

Desa Milk (500ml)

3.50


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8 Shang Pui Ting 9 Nadrah Nadhira

10 Mohd. Hasri 11 Juspinne 12 Larimi Molubi 13 Vicy Jobin 14 Allena Jupirik 15 Jessica Johnny 16 Leslie Tony 17 Vennesa Samathan 18 Eric Elton 19 Rachel Nikki 20 Heilih Luin 21 Mimi Cerolinda

22 Nurul Atika 23 Parwina Karno 24 Ching Suk Yee 25 Arty Anylia 26 Richie Justinus 27 Jelyson Philip 28 Prllchard Pitonis 29 Audrey Gee 30 Jelinah Banius 31 Betty Lee 32 Bryn Roy 33 Connie Ivy 34 Fredericka Monsun 35 Elias Augustine 36 Aaron Xavier 37 Mega Rajiman 38 Nabila Lingkan 39 Ezra Andy 40 James Laurence 41 Kerry Anne 42 Lance Lefley 43 Jansen Marshal 44 Celine Jonstus 45 Moana Marie-Lisa 46 Darren Desmond 47 Cherylene Della 48 Anette Aurora 49 Austen Bagiu 50 Jay Fung


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(b) You need to present your data to the manager. To make your data more presentable, you areadvised to include:

(i) a suitable frequency table based on your data.

(ii) suitable chart(s)/graph(s) or any relevant representations.

PIE CHART OF PREFERRED PACKAGED DRINKS BY STUDENTS

26%

20%

18%

12%

24%

Preferred Packaged Drinks by Students

Pokka Mango Milk

Dutch Lady Strawberry

Yeo's Lemon Green Tea

Marigold Orange

F&N Chrysanthemum

Types of drinks Number of students who prefer drink Pokka Mango Milk 13 Dutch Lady Strawberry 10 Yeos Lemon Green Tea 9 Marigold Orange 6 F&N Chrysanthemum 12


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BAR CHART OF PREFERRED PACKAGED DRINKS BY STUDENTS

(iii) Conclusion. -Mango Milk is the most preferred packaged drinks among the 50 students.

(c)

Based on your survey,(i) identify the sugar content in the five types of packaged drinks in (a). Use consistent

measurements so that the comparison can be done accurately. Tabulate your findings.

Types of drinks Sugar Content/250ml, x (g) x2(g)

Pokka Mango Milk 24.00 576.00

Dutch Lady Strawberry 17.50 306.25

Yeos Lemon Green Tea 23.00 529.00Marigold Orange 21.50 462.25

F&N Chrysanthemum 15.75 248.06

x = 101.75 fx =2121.56

(ii) calculate the mean and the standar deviation of the sugar content. Hence, discussthe effects of long term consumption of these packaged drinks on our health.

05

1015

13

10

6

9

12

F&N Chrysanthemum

Marigold Orange

Yeo's Lemon Green Tea

Dutch Lady Strawberry

Pokka Mango Milk


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Mean, x/N of Sugar Content = 101.75/5

= 20.35 g

Standard deviation, = [x2 /N ( )2]

= [2121.56/5 (20.35) 2]

= 424.31 414.12

= 10.19 g

Effect of long term consumption of these packaging material to our health

Blood glucose levelsSugar, because of its simpler chemical structure, may raise blood glucose levels more quicklythan starch. This finding suggests that this basic differentiation between starch and sugar isinsufficient reason to segregate these two substances for controlling blood glucose levels in diabetics,the idea behind carbohydrate counting. A more effective distinction could be that suggested bymultiple meta-studies between free sugars and naturally-occurring sugars which suggest that theyhave different impacts on health.

Obesity and Diabetes

Studies on the link between sugars and diabetes are inconclusive, with some suggesting that eatingexcessive amounts of sugar does not increase the risk of diabetes, although the extra calories fromconsuming large amounts of sugar can lead to obesity, which may itself increase the risk of diabetes. Others show correlation between refined sugar (free sugar) consumption and the onset of diabetes, and negative correlation with the consumption of fiber including a 2010 meta-analysis of eleven studies involving 310,819 participants and 15,043 cases of type 2 diabetes. This found that"SSBs (sugar-sweetened beverages) may increase the risk of metabolic syndrome and type 2 diabetesnot only through obesity but also by increasing dietary glycemic load, leading to insulin resistan ce, -cell dysfunction, and inflammation". As an overview to consumption related to chronic disease andobesity, the World Health Organization's independent meta-studies specifically distinguish free sugars("all monosaccharides and disaccharides added to foods by the manufacturer, cook or consumer, plussugars naturally present in honey, syrups and fruit juices") from sugars naturally present in food. The

reports prior to 2000 set the limits for free sugars at a maximum of 10% of carbohydrate intake,measured by energy, rather than mass, and since 2002 have aimed for a level across the entirepopulation of less than 10%. The consultation committee recognized that this goal is "controversial.However, the Consultation considered that the studies showing no effect of free sugars on excessweight have limitations."

Tooth decay

In regard to contributions to tooth decay, the role of free sugars is also recommended to be below anabsolute maximum of 10% of energy intake, with a minimum of zero. There is "convincing evidencefrom human intervention studies, epidemiological studies, animal studies and experimental studies,for an association between the amount and frequency of free sugars intake and dental caries" whileother sugars (complex carbohydrate) consumption is normally associated with a lower rate of dentalcaries.Lower rates of tooth decay have been seen in individuals wit hhereditary fructose intolerance.
http://en.wikipedia.org/wiki/Starchhttp://en.wikipedia.org/wiki/Diabetes_mellitushttp://en.wikipedia.org/wiki/Obesityhttp://en.wikipedia.org/wiki/Dietary_fiberhttp://en.wikipedia.org/wiki/Sugar-sweetened_beverage_taxhttp://en.wikipedia.org/wiki/Metabolic_syndromehttp://en.wikipedia.org/wiki/Dental_carieshttp://en.wikipedia.org/wiki/Hereditary_fructose_intolerancehttp://en.wikipedia.org/wiki/Hereditary_fructose_intolerancehttp://en.wikipedia.org/wiki/Dental_carieshttp://en.wikipedia.org/wiki/Metabolic_syndromehttp://en.wikipedia.org/wiki/Sugar-sweetened_beverage_taxhttp://en.wikipedia.org/wiki/Dietary_fiberhttp://en.wikipedia.org/wiki/Obesityhttp://en.wikipedia.org/wiki/Diabetes_mellitushttp://en.wikipedia.org/wiki/Starch


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As an environmentally friendly manufacturing company, the manager decides to minimize the usage

of the packaging material for the drinks

(a) (i) Find 10 possible dimensions of the cuboid, include the amount of packaging material used.Complete table 2.

Length (cm) Width (cm) Height (cm) Amount of packagingmaterial used (cm 2)

5.0 5.000 10 250.0005.1 4.902 10 250.0405.2 4.808 10 250.1645.3 4.717 10 250.3405.4 4.630 10 250.6045.5 4.545 10 250.8965.6 4.464 10 251.2765.7 4.386 10 251.7105.8 4.310 10 252.1965.9 4.237 10 252.736

(iii) Hence, state the dimensions of the cuboid such that the least amount of packagingmaterial is used.

Length = 5cm, width = 5 cm and height = 10cm.

(b) For aesthetic reason, the manager wants the base of the cuboid to be in golden ratio.(i) Search the internet, and write a brief explanation about the golden ratio.

In mathematics and the arts, two quantities are in the golden ratio ()if the ratio of the sum of thequantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one.

Expressed algebraically:

where the Greek letter phi ( ) represents the golden ratio. Its value is:
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Arthttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Equality_(mathematics)http://en.wikipedia.org/wiki/Phi_(letter)http://en.wikipedia.org/wiki/Phi_(letter)http://en.wikipedia.org/wiki/Equality_(mathematics)http://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Arthttp://en.wikipedia.org/wiki/Mathematics


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(ii) By using the ratio 1.6:1, find the possible dimensions of the cuboid, include the amount of packaging material used. Tabulate your answers.

Width,x (cm) Length, 1.6x (cm) Height, h = (cm) A = (cm2)

1 1.6000 156.2500 815.70002 3.2000 39.0625 419.05003 4.8000 17.3611 299.63334 6.4000 9.7656 254.32505 8.0000 6.2500 242.50006 9.6000 4.3403 250.61677 11.2000 3.1888 272.87148 12.8000 2.4414 306.36259 14.4000 1.9290 349.4778

10 16.0000 1.5625 401.250011 17.6000 1.2913 461.063612 19.2000 1.0851 528.508313 20.8000 0.9246 603.300014 22.4000 0.7972 685.235715 24.0000 0.6944 774.166716 25.6000 0.6104 869.9813

Width,x (cm) Length, 1.6x (cm) Height, h = (cm) A = (cm 2)4.1 6.5600 9.2951 251.96274.2 6.7200 8.8577 249.90044.3 6.8800 8.4505 248.12154.4 7.0400 8.0708 246.61114.5 7.2000 7.7160 245.35564.6 7.3600 7.3842 244.34244.7 7.5200 7.0733 243.56034.8 7.6800 6.7817 242.9988

4.9 7.8400 6.5077 242.64835 8.0000 6.2500 242.5000

5.1 8.1600 6.0073 242.54575.2 8.3200 5.7785 242.77805.3 8.4800 5.5625 243.18995.4 8.6400 5.3584 243.77505.5 8.8000 5.1653 244.52735.6 8.9600 4.9825 245.44135.7 9.1200 4.8092 246.51195.8 9.2800 4.6448 247.7342

5.9 9.4400 4.4887 249.1039


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Width, x (cm) Length, 1.6x (cm) Height, h = (cm) A = (cm2)

4.91 7.8560 6.4812 242.62454.92 7.8720 6.4549 242.60284.93 7.8880 6.4287 242.58304.94 7.9040 6.4027 242.56524.95 7.9200 6.3769 242.54944.96 7.9360 6.3512 242.53564.97 7.9520 6.3257 242.52384.98 7.9680 6.3003 242.51394.99 7.9840 6.2751 242.5060

5 8.0000 6.2500 242.50005.01 8.0160 6.2251 242.49605.02 8.0320 6.2003 242.49395.03 8.0480 6.1757 242.49375.04 8.0640 6.1512 242.49545.05 8.0800 6.1269 242.49915.06 8.0960 6.1027 242.50465.07 8.1120 6.0786 242.51215.08 8.1280 6.0547 242.52145.09 8.1440 6.0309 242.5326

Width,x (cm) Length, 1.6x (cm) Height, h = (cm) A = (cm2)

5.021 8.0336 6.1978 242.49385.022 8.0352 6.1954 242.49375.023 8.0368 6.1929 242.49365.024 8.0384 6.1904 242.49365.025 8.0400 6.1880 242.49355.026 8.0416 6.1855 242.49355.027 8.0432 6.1830 242.49355.028 8.0448 6.1806 242.49365.029 8.0464 6.1781 242.4936


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(iii) Hence, state the dimensions of the cuboid such that the least amount of packagingmaterial is used. Check your answer using Calculus.

Calculation using differentiation,V = 2501.6x 2h = 250h = 250/1.6x 2 A = 2[1.6x 2 + 1.6xh + xh]

= 2[1.6x 2 + 2.6xh]= 2[1.6x 2 +2.6x(250/1.6x 2)]= 3.2x 2 +812.5/x

dA/dh = 6.4x 812.5x -2 0 = 6.4x 812.5x -2

= 6.4x 3 812.5x3 = 812.5/6.4

x = 5.031.6x = 8.05h = 250/1.6(5.03) 2

= 6.18

*Length = 5.03cm, Width = 8.05cmand Height = 6.18cm.

(iv) Verify your answer in (iii) by using Geometers Sketchpad (GSP), or any ICT statisticalmethods.

2200

2000

1800

1600

1400

1200

1000

800

600

400

200

5 10 15 20

f x = 3.2 x2+812.5

x


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(c) Packaged drinks come in different sizes and volumes. Measure the dimensions of at least threedifferent types of packaged drinks. In your opinion, hive reason(s) why manufacturers use thesedimensions.

Type of drink Dimension (Length Width Height) (cm)

Reason

Milo 5.005.0010.00 Square base provides extrastability.

Ribena 4.174.0015.00 Height is more obvious providingan interesting packaging.

F&N SeasonsChrysanthemumTea

4.005.0012.50 Standard size for easy storage.

Suggest three other shapes for the packaged drinks, that will have the same height as in part3(b)(ii). Hence, discuss the pro(s) and con(s) of each of the shapes that you have chosen.Your discussion may include:

The cost of packaging material. The cost of production and storage.

Discuss briefly the usual materials used to package the drinks. Discuss which material(s)is/are more environmental friendly.

Volume:

250= ( )(r) 2(6.18)r2 = 12.871cm

= 3.587cm

Total Surface Area :r 2 r

= 2()(3.587) 2+2()(3.587)(6.18)= 80.875+139.845

= 220.720cm 2

Pro Con

- Large volume- Easy to hold - Not Stable-Not Attractive


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Volume:1/3 r 2h

250= 1/3()(r )2(6.18)r2 = 38.614

r = 6.211cm

Total Surface Area:

S = 6.2112+6.18 2 = 76.769 = 8.76cm

Surface area= (6.211)(6.211+8.76)

= 292.238cm 2

Pro Con- Stable

- Attractive- Small Volume- Hard to hold

Volume:1/3Base Areah

250 = 1/3(BaseArea)(h)

Base A = 250/2.06= 121.359

x = 121.359/2 = 11.016/2= 5.508cm

Total Surface Area:Base Area + Triangle surface

S = 6.182+5.508 2=8.278

Total area=(11.016)(11.016)+(4)(8.278)(11.016)

=486.114

Pro Con- Stable

- Attractive-Small Volume- Hard to hold

- Sharp edges and sides

Usual materials used to package drinks

Materials used for packaging consist mainly of petrochemical products, usually plastic, andalso cartons made of paperboard. Plastic packaging can be massed produced but are difficultto decompose. Thousands of years are required for plastics to break down and in the process,may release poisonous toxic, thus polluting the environment. Therefore, cartons made of paperboard are bio-degradable and more environment friendly.


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I have gained many valuable lessons throughout the project. This includes gaining knowledge on how

to carry out a good project. I realized that the planning stage is the most important part of the project

because any mistakes at this stage would jeopardize the whole assignment.

In order for the project to be successful, I collected information from the internet and brochures

pertaining to the types of packet drinks available in the market. I studied their shapes; height, width

and length. Then, based on my observation, I came up with my own design.

From another aspect, I realized the importance of sharing information with others. With reliable

friends, finding and sharing information helped me to complete the project successfully. My friends

and I discussed about the project and we shared ideas with one another. This discussion has made

me more confident to carry out future projects. Not only that, this experience has enabled me to

communicate with others effectively, thus enhancing my communication skills.

Finally, this project has taught me the importance of punctuality and carrying out proper research to

avoid plagiarism.


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Addmath 2012 (Derek's)

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