PROJECT WORK FOR ADDITIONAL MATHEMATICS 2011

Name : Muhammad Farhan Bin Sukor

Teacher’s name : Sir Mohamed Bin Masri

Class : 5 Einstein

School : Kuantan Integrated Boarding School

Introduction

Objective

We students taking Additional Mathematics are requiredto carry outa project work while we are in Form 5.This year the CurriculumDevelopment Division, Ministry of Education has prepared four tasksfor us.We are to choose and complete only ONE task based on ourarea of interest.This project can be done in groups or individually,buteach of us are expected to submit an individually written report.Uponcompletion of the Additional Mathematics Project Work,we are togain valuable experiences 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 and meaningful and hence improve their thinking skills.

Experience classroom environments where knowledge and skills are applied in meaningful ways in solving real-life

Content Page

IntroductionObjectiveHistory Of GeometryAcknowledgement

Part 1 Questions

Part 2 Questions

Part 3 Questions

Reflection

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problems.Experience classroom environments where expressing ones

mathematical thinking,reasoning and communication are highly encouraged and expected.

Experience classroom environments 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.

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.

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, to cooperate, and to share knowledge in an engaging and healthy environment.

Use technology especially the ICT appropriately andeffectively.Train ourselves to appreciate the intrinsic values of

mathematics and to become more creative and innovative.Realize the importance and the beauty of mathematics.

We are expected to submit the project work within three weeksfrom the first day the task is being administered to us.Failure tosubmit the written report will result in us not receiving certificate.

HISTORY OF GEOMETRY

         Geometry was thoroughly organized in about 300 BC, when the Greek mathematician Euclid gathered what was known at the time, added original work of his own, and arranged 465 propositions into 13 books, called 'Elements'. The books covered not only plane and solid geometry but also much of what is now known as algebra, trigonometry, and advanced arithmetic.

         Through the ages, the propositions have been rearranged, and many of the proofs are different, but the basic idea presented in the 'Elements' has not changed. In the work facts are not just cataloged but are developed in a fashionable way.

         Even in 300 BC, geometry was recognized to be not just for mathematicians. Anyone can benefit from the basic learning of geometry, which is how to follow lines of reasoning, how to say precisely what is intended, and especially how to prove basic concepts by following these lines of reasoning. Taking a course in geometry is beneficial for all students, who will find that learning to reason and prove convincingly is necessary for every profession. It is true that not everyone must prove things, but everyone is exposed to proof. Politicians, advertisers, and many other people try to offer convincing arguments. Anyone who cannot tell a good proof from a bad one may easily be persuaded in the wrong direction. Geometry provides a simplified universe, where points and lines obey believable rules and where conclusions are easily verified. By first studying how to reason in this simplified universe, people can eventually, through practice and experience, learn how to reason in a complicated world.

         Geometry in ancient times was recognized as part of everyone's education. Early Greek philosophers asked that no one come to their schools that had not learned the Elements' of Euclid. There were, and still are, many who resisted this kind of education.

         Ancient knowledge of the sciences was often wrong and wholly unsatisfactory by modern standards. However not all of the knowledge of the more learned peoples of the past was false. In fact without people like Euclid or Plato we may not have been as advanced in this age as we are. Mathematics is an adventure in ideas. Within the history of mathematics, one finds the ideas and lives of some of the most brilliant people in the history of mankind’s’ populace upon Earth. First man created a number system of base 10. Certainly, it is not just coincidence that man just so happens to have ten fingers or ten toes, for when our primitive ancestors first discovered the need to count they definitely would have used their fingers to help them along just like a child today. When primitive man learned to count up to ten he somehow differentiated him from other animals. As an object of a higher thinking, man invented ten number-sounds. The needs and possessions of primitive man were not many. When the need to count over ten aroused, he simply combined the number-sounds related with his fingers. So, if he wished to define one more than ten, he simply said one-ten. Thus our word eleven is simply a modern form of the Teutonic ein-lifon. Since those first sounds were created, man has only added five new basic number-sounds to the ten primary ones. They are “hundred,” “thousand,” “million,” “billion” (a thousand millions in America, a million millions in England), “trillion” (a million millions in America, a million-million millions in England). Because primitive man invented the same number of number-sounds as he had fingers, our number system is a decimal one, or a scale based on ten, consisting of limitless repetitions of the first ten number sounds. Undoubtedly, if nature had given man thirteen fingers instead of ten, our number system would be much changed. For instance, with a base thirteen number system we would call fifteen, two-thirteen’s. While some intelligent and well-schooled scholars might argue whether or not base ten is the most adequate number system, base ten is the irreversible favorite among all the nations. Of course, primitive man most certainly did not realize the concept of the number system he had just created. Man simply used the number-sounds loosely as adjectives. So an amount of ten fish was ten fish, whereas ten is an adjective describing the noun fish. Soon the need to keep tally on one’s counting raised. The simple solution was to make a vertical mark. Thus, on many caves we see a number of marks that the resident used to keep track of his possessions such a fish or knives. This way of record keeping is still taught today in our schools under the name of tally marks.

            The earliest continuous record of mathematical activity is from the second millennium BC when one of the few wonders of the world was created mathematics was necessary. Even the earliest Egyptian pyramid proved that the makers had a fundamental knowledge of geometry and surveying skills. The approximate time period was 2900 BC The first proof of mathematical activity in written form came about one thousand years later. The best-known sources of ancient Egyptian mathematics in written format are the Rhind Papyrus and the Moscow Papyrus. The sources provide undeniable proof that the later Egyptians had intermediate knowledge of the following mathematical problems, applications to surveying, salary distribution, calculation of area of simple geometric figures' surfaces and volumes, simple solutions for first and second degree equations. Egyptians used a base ten number system most likely because of biologic reasons (ten fingers as explained above). They used the Natural Numbers (1,2,3,4,5,6, etc.) also known as the counting numbers. The word digit, which is Latin for finger, is also another name for numbers that explains the influence of fingers upon numbers once again. The Egyptians produced a more complex system then the tally system for recording amounts. Hieroglyphs stood for groups of tens, hundreds, and thousands. The higher powers of ten made it much easier for the Egyptians

to calculate into numbers as large as one million. Our number system which is both decimal and positional (52 is not the same value as 25) differed from the Egyptian, which was additive, but not positional. The Egyptians also knew more of pi then its mere existence. They found pi to equal C/D or 4(8/9)ª whereas a equals 2. The method for ancient peoples arriving at this numerical equation was fairly easy. They simply counted how many times a string that fit the circumference of the circle fitted into the diameter, thus the rough approximation of 3. The biblical value of pi can be found in the Old Testament (I Kings vii.23 and 2 Chronicles iv.2)in the following verse “Also, he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about.” The molten sea, as we are told is round, and measures thirty cubits round about (in circumference) and ten cubits from brim to brim (in diameter). Thus the biblical value for pi is 30/10 = 3.

         Now we travel to ancient Mesopotamia, home of the early Babylonians. Unlike the Egyptians, the Babylonians developed a flexible technique for dealing with fractions. The Babylonians also succeeded in developing more sophisticated base ten arithmetic that were positional and they also stored mathematical records on clay tablets. Despite all this, the greatest and most remarkable feature of Babylonian Mathematics was their complex usage of a sexagesimal place-valued system in addition a decimal system much like our own modern one. The Babylonians counted in both groups of ten and sixty. Because of the flexibility of a sexagismal system with fractions, the Babylonians were strong in both algebra and number theory. Remaining clay tablets from the Babylonian records show solutions to first, second, and third degree equations. Also the calculations of compound interest, squares and square roots were apparent in the tablets. The sexagismal system of the Babylonians is still commonly in usage today. Our system for telling time revolves around a sexagesimal system. The same system for telling time that is used today was also used by the Babylonians. Also, we use base sixty with circles (360 degrees to a circle). Usage of the sexagesimal system was principally for economic reasons. Being, the main units of weight and money were mina,(60 shekels) and talent (60 mina). This sexagesimal arithmetic was used in commerce and in astronomy. The Babylonians used many of the more common cases of the Pythagorean Theorem for right triangles. They also used accurate formulas for solving the areas, volumes and other measurements of the easier geometric shapes as well as trapezoids. The Babylonian value for pi was a very rounded off three. Because of this crude approximation of pi, the Babylonians achieved only rough estimates of the areas of circles and other spherical, geometric objects.

         The real birth of modern math was in the era of Greece and Rome. Not only did the philosophers ask the question “how” of previous cultures, but they also asked the modern question of “why.” The goal of this new thinking was to discover and understand the reason for mans’ existence in the universe and also to find his place. The philosophers of Greece used mathematical formulas to prove propositions of mathematical properties. Some of who, like Aristotle, engaged in the theoretical study of logic and the analysis of correct reasoning. Up until this point in time, no previous culture had dealt with the negated abstract side of mathematics, of with the concept of the mathematical proof. The Greeks were interested not only in the application of mathematics but also in its philosophical significance, which was especially appreciated by Plato (429-348 BC). Plato was of the richer class of gentlemen of leisure. He, like others of his class, looked down upon the work of slaves and crafts worker. He sought relief, for the tiresome worries of life, in the study of philosophy and personal ethics. Within the walls of Plato’s academy at least three great mathematicians were taught, Theaetetus, known for the theory of irrational, Eodoxus, the theory of proportions, and also Archytas (I couldn’t find what made him great, but three books mentioned him so I will too). Indeed the motto of Plato’s academy “Let no one ignorant of geometry enter within these walls”

was fitting for the scene of the great minds who gathered here. Another great mathematician of the Greeks was Pythagoras who provided one of the first mathematical proofs and discovered incommensurable magnitudes, or irrational numbers. The Pythagorean theorem relates the sides of a right triangle with their corresponding squares. The discovery of irrational magnitudes had another consequence for the Greeks since the length of diagonals of squares could not be expressed by rational numbers in the form of A over B, the Greek number system was inadequate for describing them. As, you might have realized, without the great minds of the past our mathematical experiences would be quite different from the way they are today.

Acknowledgement

Alhamdullilah, thank you to Allah for giving the will to do my additional mathehatics project.Secondly, I would like to thank the principle of Kuantan Integrated Boarding School, Sir Awang Mohd Zaid Bin Mat Zain for giving the permission to do my Additional Mathematics Project Work. I also like to thank my Additional Mathematics teacher, Sir Mohamed Bin Masri for the guide and giving useful and important information for me to complete this project. Besides that, Iwould like to thank my parents for their support and encouragement. Lastly, special thanks to all my friends for their help and cooperation in searching for information and completing this project.

Part 1   (Find out how maths is used in cake baking and cake decorating and write about your findings)

Geometry is used to determine the suitable dimensions for the cake, to assist in designing and decorating cakes that comes in many attractive shapes and designs, to estimate volume of cake to be produced.

Calculus (differentiation) is used to determine the minimum or maximum amount of ingredients for cake-baking, to estimate min. or max. amount of cream needed for decorating, to estimate min. or max. size of cake produced.

Progressions is used to determine total weight/volume of multi-storey cakes with proportional dimensions, to estimate total ingredients needed for cake-baking, to estimate total amount of cream for decoration.

Timing calculation also conducted in the cake baking to make sure the cake is baked finely and perfectly.

Circular measure is used to calculate the area of circle for the base area of cake due to round bottom cake is given in the question.

Part 2   (bake a 5 kg round cake for your school. given the height of cake,   h   and the diameter of cake, d .)

Question 1 - Given 1 kg cake has volume 3800cm³, and   h   is 7cm, so find   d .

Volume of 5kg cake = Base area of cake x Height of cake3800 x 5 = (3.142)(d/2)² x 719000/7(3.142) = (d/2)²863.872 = (d/2)²d/2 = 29.392d = 58.784 cm

Question 2 -   Given the inner dimensions of oven: 80cm length, 60cm width, 45cm height.

Height, h (cm) Diameter, d (cm)1.0 155.532.0 109.973.0 89.794.0 77.765.0 69.556.0 63.497.0 58.788.0 54.999.0 51.8410.0 49.18

a) By doing Algebra on this formula : 3.142(d/2)²h = 19 000

b) i) State the range of heights that is NOT suitable for the cakes and explain.

h < 7 is not suitable for the cake because the resulting diameter when ‘h’ greater than 7 is greater than the width of the inner dimension of oven. So that the cake cannot be put into the oven due to its diameter is greater than the width of the inner dimension of the oven. That’s why h < 7 are not suitable for the heights of the cakes.

b) ii) Suggest and explain the most suitable dimensions (h and d) for the cake.

h = 8cm , d = 54.99cm

This is because with this height and diameter of cake, the cake can be fit perfectly into the oven. Besides, its height that is not too tall and too short is suitable for easy handling and has lower centre of gravity which prevent the cake from collapsing easily.

c) i) Form a linear equation relating d and h. Hence, plot a suitable (linear, best fit) graph based on that

equation.

3.142(d/2) ²h = 19 000(d/2) ²h = 6047.104d²h = 24 188.42d = √24 188.42/√hd = 155.53/√hlog d = log 155.53 – ½ log hlog d = - ½ log h + log 155.53

ii) By sing the graph that i have drawn, the results are : a) d when h = 10.5cm h = 10.5cm, log h = 1.021, log d = 1.680, d = 47.86cm

b) h when d = 42cm d = 42cm, log d = 1.623, log h = 1.140, h = 13.80cm

Q3)   Decorate the cake with fresh cream, with uniform thickness 1cm.

a) Estimate the amount of fresh cream needed to decorate the cake, using the dimensions

you've suggested in Q2/b/ii

Volume of cream at the top surface= Area of top surface x Height of cream= (3.142)(54.99/2)²(1)= 2375.27 cm³

Volume of cream at the side surface= Area of side surface x Height of cream= (Circumference of cake x Height of cake) x Height of cream= 2(3.142)(54.99/2)(8) x 1= 1382.23 cm³

Therefore, amount of fresh cream = 2375.27 + 1382.23 = 3757.50 cm³

b) Suggest THREE other shapes (the shape of the base of the cake) for the cake with same height (depends on the Q2/b/ii) and volume (19000cm³). Estimate the amount of fresh cream (the volume) to be used for each of those cakes.

V = 19 000cm³ , h = 8cm

1 – Rectangle-shaped base (cuboid)

19000 = base area x heightbase area = 19000/8length x width = 2375By trial and

improvement, 2375 = 50 x 47.5 (length = 50, width = 47.5, height = 8)Therefore, volume of cream= 2(Area of left/right side surface) (Height of cream) + 2(Area of front/back side surface) (Height of cream) + Vol. of top surface

= 2(8 x 50)(1) + 2(8 x 47.5)(1) + 2375 = 3935 cm³

2 – Triangle-shaped base

19000 = base area x heightbase area = 2375½ x length x width = 2375length x width = 4750By trial and improvement, 4750 = 95 x 50 (length = 95, width = 50)Slant length of triangle = √(95² + 25²)= 98.23Therefore, amount of cream= Area of rectangular front side surface(Height of cream) + 2(Area of slant rectangular left/right side surface)(Height of cream) + Vol. of top surface= (50 x 8)(1) + 2(98.23 x 8)(1) + 2375 = 4346.68 cm³

3 – Pentagon-shaped base

19000 = base area x heightbase area = 2375 = area of 5 similar isosceles triangles in a pentagontherefore:2375 = 5(length x width)475 = length x widthBy trial and improvement, 475 = 25 x 19 (length = 25, width = 19)Therefore, amount of cream= 5(area of one rectangular side surface)(height of cream) + vol. of top surface= 5(8 x 19) + 2375 = 3135 cm³

c) Based on the values above, determine the shape that require the least amount of fresh

cream to be used.

Pentagon – Shaped base/pentagonal prism is the most economical shape that can be used to create the cake since it only use the least amout of fresh cream among other shapes. So, i would say that pentagonal prism or Pentagon – Shaped base is the most suitable shape for the least omount of fresh cream to be used.

Part 3   (Find dimensions of 5kg ROUND cake (volume: 19000cm³) that require minimum amount of cream to decorate. Use two different methods, including Calculus (differentiation/integration). Also, explain whether you would choose to bake that cake with such dimensions and give reasons

Method 1: DifferentiationUse two equations for this method: the formula for volume of cake (as in Q2/a), and the formula for amount (volume) of cream to be used for the round cake (as in Q3/a).19000 = (3.142)r²h → (1)V = (3.142)r² + 2(3.142)rh → (2)From (1): h = 19000/(3.142)r² → (3)Sub. (3) into (2):V = (3.142)r² + 2(3.142)r(19000/(3.142)r²)V = (3.142)r² + (38000/r)V = (3.142)r² + 38000r-1

dV/dr = 2(3.142)r – (38000/r²)0 = 2(3.142)r – (38000/r²) -->> minimum value, therefore dV/dr = 038000/r² = 2(3.142)r38000/2(3.142) = r³6047.104 = r³r = 18.22

Sub. r = 18.22 into (3):h = 19000/(3.142)(18.22)²h = 18.22therefore, h = 18.22cm, d = 2r = 2(18.22) = 36.44cm

Method 2: Quadratic Functions

Use the two same equations as in Method 1, but only the formula for amount of cream is the main equation used as the quadratic function.Let f(r) = volume of cream, r = radius of round cake:19000 = (3.142)r²h → (1)f(r) = (3.142)r² + 2(3.142)hr → (2)From (2):f(r) = (3.142)(r² + 2hr) -->> factorize (3.142)= (3.142)[ (r + 2h/2)² – (2h/2)² ] -->> completing square, with a = (3.142), b = 2h and c = 0= (3.142)[ (r + h)² – h² ]= (3.142)(r + h)² – (3.142)h²(a = (3.142) (positive indicates min. value), min. value = f(r) = –(3.142)h², corresponding value of x = r = --h)

Sub. r = --h into (1):19000 = (3.142)(--h)²hh³ = 6047.104h = 18.22

Sub. h = 18.22 into (1):19000 = (3.142)r²(18.22)r² = 331.894r = 18.22therefore, h = 18.22 cm, d = 2r = 2(18.22) = 36.44 cmI would choose not to bake a cake with such dimensions because its dimensions are not suitable (the height is too high) and therefore less attractive. Furthermore, such cakes are difficult to be handled.

Further Exploration   (order to bake multi-storey cake)

Given that:

height, h of each cake = 6cmradius of largest cake = 31cmradius of 2 nd   cake = 10% smaller than 1 st   cake radius of 3 rd   cake = 10% smaller than 2 nd   cake...

a) Find volume of 1st, 2nd, 3rd, and 4th cakes. Determine whether the volumes form number pattern, then explain and elaborate on the number patterns.

Volume of the largest cake = 1st cake= 3.142(31)²(6) = 18 116.77cm³

Volume of the second cake= 3.142[(9/10)(31)]²(6) = 14 674.59cm³

Volume of the third cake=3.142[(9/10)(9/10)(31)]²(6) = 11 886.41cm³

Volume of the fourth cake=3.142[(9/10)(9/10)(9/10)(31)]²(6) = 9 628.00cm³

By comparing all the values obtained, it’s confirmed that the volumes of the cakes forming a number pattern

By deviding the volume of the second cake with the volume of the first cake, the resulting value is0.81 and it is remains constant for the devision of the volume of third to the volume of second cake and as well as the devision of the volume of the fourth cake to the volume of third cake. According to progression, the type of progression formed is Geometric progression where 0.81 is equal to the value of common ratio.

T2/T1 = T3/T2 = T4/T3 = 0.81

14 674.59cm³/18 116.77cm³ = 11 886.41cm³/14 674.59cm³ = 9 628.00cm³/11 886.41cm³ = 0.81

Common ratio, r = 0.81

b) Given the total mass of all the cakes should not exceed 15 kg ( total mass < 15 kg, change to volume:

total volume < 57000 cm³), find the maximum number of cakes that needs to be baked. Verify the answer using other methods.

Use Sn = (a(1 - rn)) / (1 - r), with Sn = 57000, a = 18116.772 and r = 0.81 to find n:57000 = (18116.772(1 – (0.81)n)) / (1 - 0.81)1 – 0.81n = 0.597790.40221 = 0.81n

log0.81 0.40221 = nn = log 0.40221 / log 0.81n = 4.322therefore, n ≈ 4

Verifying the answer:When n = 5:

S5 = (18116.772(1 – (0.81)5)) / (1 – 0.81) = 62104.443 > 57000 (Sn > 57000, n = 5 is not suitable)

When n = 4:S4 = (18116.772(1 – (0.81)4)) / (1 – 0.81) = 54305.767 < 57000 (Sn < 57000, n = 4 is suitable)

Okay, finally i’ve made it to complete this project 100%. At first i thought i’m not able to complete this task due to complex questions given, but when i look back the question carefully, well it’s not really complicated as i thought before. With the permission of ALLAH and with all the knowledge that Sir Mohamed has given me , finally this project is completed nicely and neatly.Hahaha...nice la sangat. Besides, this project also had taught me to be more carefull and focus while doing the complex calculations. Right now, it’s crystal clear for me that math is very useful in life. I can see that it’s used widely even in a small matter

we can used math. All i can say here is math is just awesome. The happy feelings in me cannot be described by anything. Anything except a song...This is the song that i’ve made to express my feelings about this project and MATH mainly.

Lyrics by FarhansukorGenre : JazzTitle : The Happy Holiday Destructor

LALALALA, KABOOM, KABOOM, KABOOM,I WAKE UP EVERY MORNING,I DON’T KNOW WHAT TO DO,THEN THE TV CALL ME TO WATCH IT,BUT THE ADDMATH PROJECT WORK CALL ME TO0,WHICH ONE SHOULD I HEADED TO,THIS IS SO ANNOYING I SAID TO MYSELF,THEN I SET UP A SCHEDULE FOR THE WORK TO BE DONE.

BUT GUESS WHAT ?...I DON’T FOLLOW THE SCHEDULE,MY HOLIDAY IS DESTRUCTED AGAIN,OH NO, NO, NO.MY HOLIDAY IS DESTRUCTED AGAIN,OH NO, NO,NO.

TO PREVENT THIS FROM KEEP HAPPENING,I GO THROUH THE ADD MATH PROJECT AND I DO IT WITH AN ANNOYING FEELING,FOR THE FIRST ONE HOUR DOING IT,I JUST WANT TO DIE,WHEN IT EXCEED 2 HOURS,I LAUGH AND FEEL EXCITED TO DO IT AND I WANNA CRY.

ADDMATH PROJECT WORK IS FUN,AND I THINK I WANNA SHOUT IN THE AIR,ADD MATH PROJECT WORK IS FUN FUN FUN,EVEN THOUGH IT DESTRUCTS MY HOLIDAY.BUT IT JUST FUN..TUP TUP PABADIDAB........

THE END