CONTENTIntroductionAppreciationHistory of StatisticPart 1Part 2Part 3Further ExplorationConclusionReflectionReferences

1

IntroductionThis project is carried out by every student who taking Additional Mathematic in their SPM examination. This project carries such aims:-

·        Apply and adapt a variety of problem solving strategies to solve routine and non-routine problem·        Experience classroom environments which are challenging, interesting and meaningful and hence improve their thinking skills·        Experience classroom environments where knowledge and skills are applied in meaningful ways in solving real-life problems·        Experience classroom environments where expressing ones mathematical thinking, reasoning and communication are highly encouraged and expected·        Experience classroom environment that stimulates and enhances effective learning.·        Acquire effective mathematical communication through oral and writing, and to use the language of mathematics to express mathematical ideas correctly and precisely

2

·        Enhance acquisition of mathematical knowledge and skills through problem-solving in ways that increase interest and confidence·        Prepare ourselves for the demand of our future undertakings and in workplace·        Realize 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, to cooperate, 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 mathematics and to become more creative and innovative·        Realize the importance and the beauty of mathematics.

3

AwcknowledgeFirst of all, I would like to give thanks to God for giving me strength to get this project done on time.My gratifications are also for my parents who never fail to help me to finish this project. They always encourage me to complete this project so I will not procrastinate in doing it. They also provides me everything that I need in doing this project such as money, computer, paper , Internet ,etc.Not forgotten, my Additional Mathematics teacher, Sir Erwin Balanting for guiding me and my friends throughout this project. Last but not least, I would like to thanks my friends who were doing this project together with me, Excyvera Kuyun , Vellendra Andrew, Vannesa Jeffry, Catherine Lojamin and Cheryl Shenie. They were helpful and supportive. We discussed, shared and combined our ideas and had our task done. Thank you so much !

4

History of StatisticBy the 18th century, the term "statistics" designated the systematic collection of demographic and economic data by states. In the early 19th century, the meaning of "statistics" broadened, then including the discipline concerned with the collection, summary, and analysis of data. Today statistics is widely employed in government, business, and all the sciences. Electronic computers have expedited statistical computation, and have allowed statisticians to develop "computer-intensive" methods. The term "mathematical statistics" designates the mathematical theories of probability and statistical inference, which are used in statistical practice. The relation between statistics and probability theory developed rather late, however. In the 19th century, statistics increasingly used probability theory, whose initial results were found in the17th and18th centuries, particularly in the analysis of games of chance(gambling). By 1800,astronomy used probability models and statistical theories, particularly the method of least squares, which was invented by Legendre and Gauss Early probability theory and statistics was systematized and extended by Laplace; following Laplace, probability and statistics have been in continual development. In the19th century, social scientists used statistical reasoning and probability models to advance the new sciences of experimental psychology and sociology; physical scientists

5

used statistical reasoning and probability models to advance the new sciences of thermodynamics and statistical mechanics. The development of statistic was closely associated with the development of inductive logic and the scientific method Statistics is not a field of mathematics but an autonomous mathematical science, like computer science or operations research. Unlike mathematics, statistics had its origins in public administration and maintains a special concern with demography and economics. Being concerned with the scientific method and inductive logic, statistical theory has close association with the philosophy of science; with its emphasis on learning from data and making best predictions, statistics has great overlap with the decision science and microeconomics. With its concerns with data, statistics has overlap with information science and computer science.

6

Statistic Today During the 20th century, the creation of precise instruments for agricultural research, public health concerns (epidemiology, biostatistics, etc.), industrial quality control, and economic and social purposes (unemployment rate, econometrics, etc.) necessitated substantial advances in statistical practices. Today the used of statistic has broadened far beyond its origin. Individuals and organizations use statistics to understand data and make informed decisions throughout the natural and social sciences, medicines, business, and other area. Statistics are generally regarded not as the subfield of mathematics but rather as a distinct, allied, field. Many universities maintain separate mathematics and statistic departments. Statistic is also taught in department as diverse as psychology , education and public health.

7

Part 1(a)  Types of Packaged Drinks

   Brand   ManufacturerYeo’s Yeo Hiap Seng (M) Sdn.BhdDrinho        Lam Soon (M) Sd.BhdNestle Nestle Products Sdn.BhdSusu Segar Desa Desa Cattle (Sabah) Sdn. BhdMarigold Malaysia Milk Ind (M) Sdn.Bhd

b)

RM 1.30

Location: Anak Mami Superstore

8

                                                                            RM 1.30Location : Survey Supermarket

RM 1.60Location : Jasmine Superstore

9

RM 4.50Location :Desa Fresh Mart

RM 1.20Location : Giant Supermarket

10

Part 2(a)

N0. Student’s Name Yeo’s Marigold Nestle Drinho Desa1 Achibal Akbal       2 Aeris Justin       3 Alfera Dusing        4 Allan Edward Muyuk       5 Allister Albert       6 Amanda Eva       7 Amarina Suid        8 Aqmal Salleh       9 Azrul Nizam Yucque       10 Brenda Christie       11 Caroline Eveth 12 Catherine Lojamin        13 Cheryl Shenie Senleh        14 Clement Paul       15 Cody Legge        16 Edna Esther Jeffery       17 Emy Pip        18 Excyvera Kuyun       19 Irsyad Hamilin        20 Issac Roland        21 Jaeronne Japius        22 Jasmine Bruner       23 Jessica Tompiris        24 Jezreel Pontilas       25 Kado Ukun              26 Khaireez Azizi       27 Krista Ojingkil        28 Lionel Tambur        29 Luke Anselm        30 Marlane Peter Ringgingon       31 Marylyn Joniwod        32 Maxwell Missun       33 Nor Syafika Zubaidi       34 Palingo Gedion       35 Rachell John       36 Rick Falco Polodius       37 Sharron Roland       

11

38 Shasha Bernard       39 Shermaine Julang       40 Sherman Dawayan        41 Suezanna Jamal        42 Suzie Elyse Aon       43 Sylvester Jeffery        44 Tiffany Gontol       45 Tress Faustina Jeffery        46 Vannesa Jeffry       47 Vellendra Glenda Andrew        48 Venella Sandah       49 Veraneca Kinatu       50 Vincent Podimun       

                                                 

(b)  (i)    

Brand Number Of StudentsYeo’s 6

Marigold 9Nestle 10Drinho 9Desa 16

(ii)

12

Yeo's Marygold Nestle Drinho Desa0

2

4

6

8

10

12

14

16

18

Frequency of students

Frequency of students

(iii) Through the study of 50 students, Desa is a very popular drink with a total of 16 votes and was rated the highest among other drinks. Meanwhile, Nestle is the preferred beverage with 10 votes, Marigold and Drinho is the third drink favorite by the students with same 10 votes. Finally, YEO `S has received the least votes and is drinking less favorite by the students with 6 votes.

13

(c) (i)    

Type Of Brand Sugar ContentYeo’s 23gDrinho 18.5gNestle 7g

Marigold 11.0gDesa 6g

(ii)  Mean : 23+19+7+11+6 = 66                                       =66/5                                       =13.2Standard Deviation :  .

σ  = -(174.24)   =12.20

(iii) Effect of long time consumption of packaged drinks on our health.  a) Tooth Decay: All soft drinks are acidic which corrodes the teeth by eroding its enamel. The high amount of sugar consumed through soft drinks lead to the development of bacteria that attack the teeth thus aggravating dental problems. People who drink three or more glasses of aerated drinks (gaseous drinks) daily have much higher chances of dental decay, fillings and teeth loss. Therefore, soft drinks contain acid and sugar that corrode and destroy the teeth in one shot.b) Obesity: Soft drinks are mainly composed of filtered H2O, artificial additives and refined sugar. Thus, they lack nutritional value and only add up calories through their refined sugar; therefore, they make you gain weight. But is shifting to "Diet Soft Drink" the solution? The fact is that diet soft drinks contain aspartame (an artificial low-calorie sweetener). Although aspartame does not add up the calories it makes you feel hungrier and crave for food. The Food and Drug Administration (FDA) no longer allows foods containing aspartame to be labeled as "weight reduction product ". Now it insists to label these products simply as "Diet Drink" or "Diet Food ". Research also shows that aspartame causes migraines, dizziness and more over it reduces your memoryIt has been experimentally proved that soft drinks are one of the prominent reasons for obesity. 14

The weight gain is directly related to the amount of soft drink that a person takes in. With every single can, people unknowingly add many extra calories to their body weight.

c) Diabetes :Another very common effect of soft drinks is diabetes. With every can of soft drink, people add large amount of sugar in their body. Long habitual intake of soft drinks can lead to ineffective insulin production from pancreases which consequently affect the sugar level in the body. This further leads to diabetes.d) Osteoporosis: It has been proved that frequent consumption of soft drink can lead to weakening of the bones and osteoporosis. Soft drinks impair the calcification of the growing bones in children.e)Kidney stones: his fact may shock you, but is quite true. Researches have proved that constant and habitual intake of soft drink can lead to kidney stones formation. This happens because of the acidic and mineral radical balance. The body tries to buffer the acidity caused by the soft drinks with the calcium from the

bones. This leads to calcium erosion, which ultimately gets settled in the kidney in the form of stones.

f) Indigesdtion:Soft drink also leads to impaired digestive system. Soft drink contains phosphoric acid which competes with the hydrochloric acid present in stomach and affects its functioning. The ineffectiveness of stomach leads to undigested food which further causes indigestion and gassiness.

g) Dry throat :Soft drinks cause dehydration in the body. Both the sugar and caffeine components of soft drinks are dehydrating agents. They both lead to excess urination, which makes you thirstier than before.

h) High blood pressure: Soft drinks have strong caffeine content. Caffeine causes irritability, restlessness, tension, high blood pressure, excessive urination and other side effects. It is also believed that soft drinks increase blood pressure. Some studies also claim that soft drink has harmful effects on liver.

The sweetener used in soft drinks is Aspartame. It is 200 times sweeter than normal sugar and is far more harmful with many side effects.Next time, when head towards the refrigerator or a store

15

to satiate your thirst with a can of soft drink, make sure you remind yourself of all the side effects that gulping down the beverage will bring forth.

16

 Part 3(a)(i)Length (cm) Width(cm) Height(cm)

Amount Packaging Material Used (cm)

4 6 10.399980 255.9996004 6 10.399990 255.9998004 6 10.399995 255.9999004 6 10.399911 255.9982204 6 10.416667 256.3333404 6 10.416669 256.3333804 6 10.416711 256.3342204 6 10.417100 256.3420004 6 10.418967 256.3793444 6 10.418950 256.379000

(a)(ii)

Length, L (cm) Width, W (cm) Height, H (cm)

Total Surface Area (or

Amount of Packaging

Material Used), A (cm2)

Volume of Cuboid, V (cm3)

4 6 10.416667 256.333340 250.0000084 6 10 256 250

17

(b)(i) Golden RatioThe golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years. According to Mario Livio: Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. The division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons. The Greeksusually attributed discovery of this concept to Pythagoras or his followers. The regular pentagram, which has a regular pentagon inscribed within it, was the Pythagoreans' symbol.Euclid's Elements (Greek: Στοιχεῖα) provides the first known written definition of what is now called the golden ratio: "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less." Euclid explains a construction for cutting (sectioning) a line "in extreme and mean ratio", i.e. the golden ratio. Throughout the Elements, several propositions (theorems in modern terminology) and their proofs employ the golden ratio. Some of these propositions show that the golden ratio is an irrational number. The name "extreme and mean ratio" was the principal term used from the 3rd century BC until about the 18th century. The modern history of the golden ratio starts with Luca Pacioli's De divina proportione of 1509, which captured the imagination [citation needed] of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of the golden ratio. Michael Maestlin, first to publish a decimal approximation of the golden ratio, in 1597.The first known approximation of the (inverse) golden ratio by a decimal fraction, stated as "about 0.6180340," was written in 1597 by Michael Maestlin of the University of Tübingen in a letter to his former student Johannes Kepler. Since the twentieth century, the golden ratio has been represented by the Greek letter Φ or φ (phi, after Phidias, a sculptor who is said to have employed it) or less commonly by τ (tau, the first letter of the ancient Greek root τομή—meaning cut).

18

(b)(ii)

Width,x (cm) Length, 1.6x (cm) Height,h (cm) Total surface area,A

(cm2)

1 1.6 156.250000 815.7000002 3.2 39.062500 419.0500003 4.8 17.361111 299.6333324 6.4 9.765625 254.3250005 8 6.250000 242.5000006 9.6 4.340278 250.6166747 11.2 3.188776 272.8714468 12.8 2.441406 306.3624909 14.4 1.929012 349.47776210 16 1.562500 401.250000

(b)(iii)

Width, x (cm)

Length, 1.6x (cm) Height, h (cm)Total Surface Area ,

A (cm2)Volume of Cuboid, V

(cm3)5 8 6.250000 242.500000 250.000000

Calculation using differentiation, V = 250 cm31.6x²h=250h=250/1.6x²A=2[1.6x² + 1.6xh+xh]  =2[1.6x² + 2.6xh]  =2[1.6x² + 2.6x (250/1.6x²)]  =3.2x² + 812.5/xDy/dx= 6.4x – 812.5xˉ²0 = 6.4x – 812.5/x²0 = 6.4x³ - 812.5x³ = 812.5/6.4x = 5.03 cm ( from tabulation table above, x = 5 cm; ok)1.6x = 8.0 cm

(b)(iv)H=250/1.6(5.)² = 6.25 cm .:  Length = 8.0cm;  Width = 5.0cm; and Height = 6.25cm

19

h = 6.25 cm (from tabulation table, b (iii))1/3 x 6.25x² = 250X = 10.954 cm√6.25²+(10.954/2)² = 8.3102Total surface area = ½(10.954 x 8.3102) x 4 + 10.954² = 301.99cm²

Note : Total surface area for pyramid > total surface area for cuboid packaged drinks; more material are needed : not good

(c)

h = 6.25 cm (from tabulation table, b (iii))1/3 Π r² x 6.25= 250r=6.18 cm√6.25² + (6.18/2) ² = 6.972Total surface area = Πrl + Πr² = Π(6.18)(6.972) + Π(6.18)² = 255.3466cm²

20

h = 6.25 cm (from tabulation table, b (iii))Πr² x 6.25 = 250r = 3.568Total surface area= 2Π(3.568)² + 2Π(3.568)(6.25) = 220.1cm²

FURTHER EXPLORATION(a)

Shape s of packaging Pro(s) Con(s)

21

Pyramid -the stability of the drink packaging will be more stable because the base area larger-the shape of the drink packaging become more unique because it’s rarely found in the market.

-the cost of packaging material will be higher because more total surface area required.

Cone -the stability of the drink packaging will be more stable because the base area larger-the shape of the drink packaging become more unique because it’s rarely found in the market.

-the cost of packaging material will be higher because more total surface area required.

Cyclinder -the drink packaging has clean line construction and easy to clean, disinfect and has smooth lines.- the shape of packaging is usually found in the market.

-the cost of packaging material is lower because it has small total surface.

22

Futher Exploration

Material usually used to packaged drink.

Usually drinks are packaged using materials like plastic, aluminum, and glass in different shapes and form. It usually come packaged in bottle ( plastic or glass ) and can ( aluminum ) .

Plastic

Composition of Plastic

Most organic polymers are plastics. The vast majority of these polymers are based on chains of carbon atoms alone or with oxygen, sulfur, or nitrogen as well. The backbone is that part of the chain on the main "path" linking a large number of repeat units together. To customize the properties of a plastic, different molecular groups "hang" from the backbone (usually they are "hung" as part of the monomers before linking monomers together to form the polymer chain). The structure of these "side chains" influence the properties of the polymer. This fine tuning of the properties of the polymer by repeating unit's molecular structure has allowed plastics to become an indispensable part of the twenty-first century world

Additives of plastic

. Most plastics contain other organic or inorganic compounds blended in. The amount of additives ranges from zero percentage for polymers used to wrap foods to more than 50% for certain electronic applications. The average content of additives is 20% by weight of the polymer. Fillers improve performance and/or reduce production costs. Stabilizing additives include fire retardants to lower the flammability of the material. Many plastics contain fillers, relatively inert and inexpensive materials that make the product cheaper by weight. Typically fillers are mineral in origin, e.g., chalk. Some fillers are more chemically active and are called reinforcing agents. Since many organic polymers are too rigid for particular applications, they are blended with plasticizers, oily compounds that confer improved rheology. Colorants are of course common additives, although their weight contribution is small. Many of the controversies associated with plastics are associated with the additives.

Recycling of Plastic

Plastic can be recycled. There are about 50 types of plastic used in the industry. The plastics industry has developed a coding system to help consumers identify different types of plastic resins used in packaging. The codes can be found on the bottom of most plastic containers. Plastic Bottles: The material used to make plastic soda bottles (polythylene terephthalate or PET) is recyclable. These bottles are coded with the number 1 and the letters PETE. About 30 percent of all PET bottles sold are recycled into a variety of products.

Glass

Composition of Glass

Most glass is a mixture of silica obtained from beds of fine sand or from pulverized sandstone; an alkali to lower the melting point, usually a form of soda or, for finer glass, potash; lime as a stabilizer; and cullet (waste glass) to assist in melting the mixture. The properties of glass are varied by adding other substances, commonly in the form of oxides, e.g., lead, for brilliance and weight; boron, for thermal and electrical resistance; barium, to increase the refractive index, as in optical glass; cerium, to absorb infrared rays; metallic oxides, to impart color; and manganese, for decolorizing. The term “crystal glass,” derived from rock crystal, was at first applied to clear, highly refractive glass; it has come to denote in the trade a high-grade, colorless glass and is sometimes applied to any fine hand-blown glass.

23

Additivities of Glass

Glass also contains additives. Additives are used in glass coloring and color marking. Glass coloring and color marking may be obtained by addition of coloring ions, precipitation of nanometer sized colloides (so-called striking glasses such as "ruby gold" or red "selenium ruby" by colored inclusions (as in milk glass and smoked glass), by light scattering (as in phase separated glass), by dichroic coatings (seedichroic glass), or by colored coatings. The additives used in glass will not affect the user.

Recycling of Glass

Glass is endlessly recyclable, and most glass bottles and jars produced contain at least 25% recycled glass – which also requires 75% less energy to produce than glass made from new materials. In recycling glass . the glass is sorted by colour and washed to remove impurities The glass is then crushed and melted, then moulded into new products such as bottles and jars. Or it may be used for alternative purposes such as brick manufacture or decorative uses. The glass is then sent back to the shops ready to be used again.

Aluminum

Composition of aluminum can

Aluminum is a silvery white member of the boron group of chemical elements. It has the symbol Al, and its atomic number is 13. It is not soluble in water under normal circumstances. Aluminum metal is so chemically reactive that native specimens are rare and limited to extreme reducing environments. Instead, it is found combined in over 270 different minerals.[4] The chief ore of aluminum is bauxite. It is remarkable for the metal's low density and for its ability to resist corrosion due to the phenomenon of passivation. 

Additives of aluminum can

There are no additives in aluminum can .

Recycling of aluminum can

Aluminum is a sustainable metal and can be recycled over and over again. A can is generally turned into a new can and back on store shelves within 60 days. Cans are usually available through curbside pickup or community drop-off

locations nationwide.

24

Eco-friendly Material for pakaging drinks

The term “eco-friendly” is used to describe activities which are good for the environment. It is a shortening of “ecologically friendly,” and you may also hear terms like “environmentally friendly” or “green” used to describe similar activities. There are a range of ways in which activities can be eco-friendly, ranging from products which are constructed in an environmentally friendly way to making lifestyle changes which are designed to benefit the environment. The most eco-friendly material for packaging drinks is glass.

Glass is more environmentally friendly because it doesn't use fossil fuels to produce, however as far as recycling is concerned it is difficult to say. Conclusive life-cycle studies are yet to be fully analyzed. However glass can be more easily returned to its natural state of composite sand. Whereas plastic takes on average 2000 years to biodegrade. Meanwhile, aluminum can also be recycle. But it cannot be reuse again like glass. It has to undergo several process in the treatment plant to be able to be used again. This require more energy .

On the other hand, glass is more enviromentally friendly because glass could be used over and over again. It is long lasting and Glass is 100% recyclable and can be recycled endlessly with no loss in quality

or purity. Glass recycling is a closed loop system, creating no additional waste or by-products. Glass is a mono-material, meaning it is not made up of several different materials.

25

.

Plastic bottle use in packaging drinks.

Glass bottle used in packaging drinks.

Aluminum can used in packaging drinks.

26

Conclusion After doing research, answering questions, drawing graphs and some problem solving, I saw that the usage of statistics is important in daily life. It is not just widely used in markets but also in interpreting the condition of the surrounding like the air or the water. Especially in conducting an air-pollution survey. In conclusion, statistics is a daily life necessities. Without it, surveys can’t be conducted, the stock market can’t be interpret and many more. So, we should be thankful of the people who contribute in the idea of statistics.

27

ReflectionWhile I conducting this project, a lot of information that I found. I

have learnt how statistics appear in our daily life.

Apart from that, this project encourages the student to work together and share their knowledge. It is also encourage student to gather information from the internet, improve thinking skills and promote effective mathematical communication.

Not only that, I had learned some moral values that I practice. This project had taught me to responsible on the works that are given to me to be completed. This project also had made me felt more confidence to do works and not to give easily when we could not find the solution for the question. I also learned to be more discipline on time, which I was given about 2 weeks to complete this project and pass up to my teacher just in time. I also enjoy doing this project I spend my time with friends to complete this project and it had bonded up our friendship deeply.

Last but not least, I proposed this project should be continue because it brings a lot of moral value to the student and also test the students understanding in Additional Mathematics.

28

Confident

Cooperation

29

Friendship - bonding

Discipline

30

Enhance Mathematical skill

31