kerja kursus additional mathematics 2011

8/4/2019 Kerja Kursus Additional Mathematics 2011

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MATHEMATICS IN

BAKING ANDDECORATING

CAKE

2011

Name : Muhammad Muizzuddin Bin NazriClass : 506

I/C Number : 941130-06-5203Teacher : Puan Saadiah binti Abu Bakar


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CONTENTS

Cover Page1

Contents...2

Appreciation.3

Introduction4 - 9

Conjectures10 - 11

Problem Statement12 - 26

Further Exploration27 - 30

Conclusion.31

Reflection32 - 35

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APPRECIATION

Alhamdulillah, finally I can complete this Additional Mathematics Project Work.

Here I want to give my appreciation to all who have helped me in completing this project

work.

Firstly, I would like to thank my parents, Encik Nazri Bin Haji Pendek and Puan

Nik Noriza Binti Hamzah for their moral support to me. Especially my father who gave

me the opportunity to complete the project using a computer. Secondly, I want to thank

my Mathematics teacher, Puan Hajah Sa'adiah Binti Abu Bakar as much guiding me in

completing this project work. To my helpful friends of Al-Khawarizmi, 506, my infinite

gratitude goes to all of you for the assistance in completing the project work. May we

together get 11 A + in SPM 2011. Amin. Next lot of thanks and gratitude to fellow

friends, form five batches of 56, Kolej Islam Sultan Alam Shah, because provides many

up-to-date informations for the Additional Mathematics Project Work. May we together

get 11 A + in SPM 2011 and be the best batch in Malaysia for SPM 2011. Amin.

Finally, I want to thank those involved directly or indirectly in my efforts to

complete this project work. May Allah bless you all. Thank you.

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INTRODUCTION

1. History of geometry

Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid'sElements

The earliest recorded beginnings of

geometry can be traced to ancient Mesopotamia,

Egypt, and the Indus Valley from around 3000 BCE. Early geometry was a collection of

empirically discovered principles concerning lengths, angles, areas, and volumes, which

were developed to meet some practical need in surveying, construction, astronomy, and

various crafts. The earliest known texts on geometry are the EgyptianRhind Papyrus and

Moscow Papyrus, the Babylonian clay tablets, and the IndianShulba Sutras, while the

Chinese had the work ofMozi, Zhang Heng, and theNine Chapters on the Mathematical

Art, edited by Liu Hui. South of Egypt the ancient Nubians established a system of

geometry including early versions of sun clocks.

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http://en.wikipedia.org/wiki/File:Woman_teaching_geometry.jpghttp://en.wikipedia.org/wiki/File:Woman_teaching_geometry.jpghttp://en.wikipedia.org/wiki/File:Woman_teaching_geometry.jpghttp://en.wikipedia.org/wiki/File:Woman_teaching_geometry.jpghttp://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Mesopotamiahttp://en.wikipedia.org/wiki/Ancient_Egypthttp://en.wikipedia.org/wiki/Indus_Valley_Civilizationhttp://en.wikipedia.org/wiki/Surveyinghttp://en.wikipedia.org/wiki/Constructionhttp://en.wikipedia.org/wiki/Astronomyhttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Babylonian_mathematicshttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Shulba_Sutrashttp://en.wikipedia.org/wiki/Mozihttp://en.wikipedia.org/wiki/Zhang_Henghttp://en.wikipedia.org/wiki/Nine_Chapters_on_the_Mathematical_Arthttp://en.wikipedia.org/wiki/Nine_Chapters_on_the_Mathematical_Arthttp://en.wikipedia.org/wiki/Liu_Huihttp://en.wikipedia.org/wiki/Nubiahttp://en.wikipedia.org/wiki/File:Woman_teaching_geometry.jpghttp://en.wikipedia.org/wiki/File:Woman_teaching_geometry.jpghttp://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Mesopotamiahttp://en.wikipedia.org/wiki/Ancient_Egypthttp://en.wikipedia.org/wiki/Indus_Valley_Civilizationhttp://en.wikipedia.org/wiki/Surveyinghttp://en.wikipedia.org/wiki/Constructionhttp://en.wikipedia.org/wiki/Astronomyhttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Babylonian_mathematicshttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Shulba_Sutrashttp://en.wikipedia.org/wiki/Mozihttp://en.wikipedia.org/wiki/Zhang_Henghttp://en.wikipedia.org/wiki/Nine_Chapters_on_the_Mathematical_Arthttp://en.wikipedia.org/wiki/Nine_Chapters_on_the_Mathematical_Arthttp://en.wikipedia.org/wiki/Liu_Huihttp://en.wikipedia.org/wiki/Nubia


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Until relatively recently (i.e. the last 200 years), the teaching and development of

geometry in Europe and the Islamic world was based on Greek geometry. Euclid's

Elements (c. 300 BCE) was one of the most important early texts on geometry, in which

he presented geometry in an ideal axiomatic form, which came to be known as Euclidean

geometry. The treatise is not, as is sometimes thought, a compendium of all that

Hellenistic mathematicians knew about geometry at that time; rather, it is an elementary

introduction to it; Euclid himself wrote eight more advanced books on geometry. We

know from other references that Euclids was not the first elementary geometry textbook,

but the others fell into disuse and were lost.

In the Middle Ages, mathematics in medieval Islam contributed to the

development of geometry, especially algebraic geometry and geometric algebra. Al-

Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating

the cube to problems in algebra. Thbit ibn Qurra (known as Thebit in Latin) (836901)

dealt with arithmetical operations applied to ratios of geometrical quantities, andcontributed to the development ofanalytic geometry. Omar Khayym (10481131) found

geometric solutions to cubic equations, and his extensive studies of theparallel postulate

contributed to the development ofnon-Euclidian geometry.

The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam

andNasir al-Din al-Tusi on quadrilaterals, including the Lambert

quadrilateral and Saccheri quadrilateral, were the first theorems

on elliptical geometry and hyperbolic geometry, and along with

their alternative postulates, such as Playfair's axiom, these works

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http://en.wikipedia.org/wiki/History_of_geometry#Greek_geometryhttp://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Axiomhttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Hellenistichttp://en.wikipedia.org/wiki/Middle_Ageshttp://en.wikipedia.org/wiki/Mathematics_in_medieval_Islamhttp://en.wikipedia.org/wiki/Algebraic_geometryhttp://en.wikipedia.org/wiki/Geometric_algebrahttp://en.wikipedia.org/wiki/Al-Mahanihttp://en.wikipedia.org/wiki/Al-Mahanihttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Th%C4%81bit_ibn_Qurrahttp://en.wikipedia.org/wiki/Latinhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Analytic_geometryhttp://en.wikipedia.org/wiki/Omar_Khayy%C3%A1mhttp://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Parallel_postulatehttp://en.wikipedia.org/wiki/Non-Euclidian_geometryhttp://en.wikipedia.org/wiki/Non-Euclidian_geometryhttp://en.wikipedia.org/wiki/Ibn_al-Haythamhttp://en.wikipedia.org/wiki/Ibn_al-Haythamhttp://en.wikipedia.org/wiki/Nasir_al-Din_al-Tusihttp://en.wikipedia.org/wiki/Quadrilateralhttp://en.wikipedia.org/wiki/Lambert_quadrilateralhttp://en.wikipedia.org/wiki/Lambert_quadrilateralhttp://en.wikipedia.org/wiki/Saccheri_quadrilateralhttp://en.wikipedia.org/wiki/Elliptical_geometryhttp://en.wikipedia.org/wiki/Hyperbolic_geometryhttp://en.wikipedia.org/wiki/Playfair's_axiomhttp://en.wikipedia.org/wiki/History_of_geometry#Greek_geometryhttp://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Axiomhttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Hellenistichttp://en.wikipedia.org/wiki/Middle_Ageshttp://en.wikipedia.org/wiki/Mathematics_in_medieval_Islamhttp://en.wikipedia.org/wiki/Algebraic_geometryhttp://en.wikipedia.org/wiki/Geometric_algebrahttp://en.wikipedia.org/wiki/Al-Mahanihttp://en.wikipedia.org/wiki/Al-Mahanihttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Th%C4%81bit_ibn_Qurrahttp://en.wikipedia.org/wiki/Latinhttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Analytic_geometryhttp://en.wikipedia.org/wiki/Omar_Khayy%C3%A1mhttp://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Parallel_postulatehttp://en.wikipedia.org/wiki/Non-Euclidian_geometryhttp://en.wikipedia.org/wiki/Ibn_al-Haythamhttp://en.wikipedia.org/wiki/Nasir_al-Din_al-Tusihttp://en.wikipedia.org/wiki/Quadrilateralhttp://en.wikipedia.org/wiki/Lambert_quadrilateralhttp://en.wikipedia.org/wiki/Lambert_quadrilateralhttp://en.wikipedia.org/wiki/Saccheri_quadrilateralhttp://en.wikipedia.org/wiki/Elliptical_geometryhttp://en.wikipedia.org/wiki/Hyperbolic_geometryhttp://en.wikipedia.org/wiki/Playfair's_axiom


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had a considerable influence on the development of non-Euclidean geometry among later

European geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis, and

Giovanni Girolamo Saccheri.

In the early 17th century, there were two important

developments in geometry. The first, and most important,

was the creation of analytic geometry, or geometry with

coordinates and equations, by Ren Descartes (1596

1650) and Pierre de Fermat (16011665). This was a

necessary precursor to the development ofcalculus and a

precise quantitative science ofphysics. The second

geometric development of this period was the systematic study ofprojective geometry by

Girard Desargues (15911661). Projective geometry is the study of geometry without

measurement, just the study of how points align with each other.

Two developments in geometry in the 19th century changed the way it had been

studied previously. These were the discovery of non-Euclidean geometries by

Lobachevsky, Bolyai and Gauss and of the formulation of symmetry as the central

consideration in the Erlangen Programme of Felix Klein (which generalized the

Euclidean and non Euclidean geometries). Two of the master geometers of the time were

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http://en.wikipedia.org/wiki/Witelohttp://en.wikipedia.org/wiki/Levi_ben_Gersonhttp://en.wikipedia.org/wiki/Alfonsohttp://en.wikipedia.org/wiki/John_Wallishttp://en.wikipedia.org/wiki/Giovanni_Girolamo_Saccherihttp://en.wikipedia.org/wiki/Coordinate_systemhttp://en.wikipedia.org/wiki/Equationshttp://en.wikipedia.org/wiki/Equationshttp://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Projective_geometryhttp://en.wikipedia.org/wiki/Girard_Desargueshttp://en.wikipedia.org/wiki/Non-Euclidean_geometryhttp://en.wikipedia.org/wiki/Nikolai_Ivanovich_Lobachevskyhttp://en.wikipedia.org/wiki/J%C3%A1nos_Bolyaihttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Symmetryhttp://en.wikipedia.org/wiki/Erlangen_Programmehttp://en.wikipedia.org/wiki/Erlangen_Programmehttp://en.wikipedia.org/wiki/Felix_Kleinhttp://en.wikipedia.org/wiki/Witelohttp://en.wikipedia.org/wiki/Levi_ben_Gersonhttp://en.wikipedia.org/wiki/Alfonsohttp://en.wikipedia.org/wiki/John_Wallishttp://en.wikipedia.org/wiki/Giovanni_Girolamo_Saccherihttp://en.wikipedia.org/wiki/Coordinate_systemhttp://en.wikipedia.org/wiki/Equationshttp://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Projective_geometryhttp://en.wikipedia.org/wiki/Girard_Desargueshttp://en.wikipedia.org/wiki/Non-Euclidean_geometryhttp://en.wikipedia.org/wiki/Nikolai_Ivanovich_Lobachevskyhttp://en.wikipedia.org/wiki/J%C3%A1nos_Bolyaihttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Symmetryhttp://en.wikipedia.org/wiki/Erlangen_Programmehttp://en.wikipedia.org/wiki/Felix_Klein


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Bernhard Riemann, working primarily with tools from mathematical analysis, and

introducing the Riemann surface, and Henri Poincar, the founder ofalgebraic topology

and the geometric theory ofdynamical systems. As a consequence of these major changes

in the conception of geometry, the concept of "space" became something rich and varied,

and the natural background for theories as different as complex analysis and classical

mechanics.

1. History of Calculus(Differentiation)

The concept of a derivative in the sense of a tangent line is a very old one, familiar to

Greek geometers such as Euclid (c. 300 BC), Archimedes (c. 287212 BC) and

Apollonius of Perga (c. 262190 BC). Archimedes also introduced the use of

infinitesimals, although these were primarily used to study areas and volumes rather than

derivatives and tangents; see Archimedes' use of infinitesimals.

The use of infinitesimals to study rates of change can be found in Indian

mathematics, perhaps as early as 500 AD, when the astronomer and mathematician

Aryabhata (476550) used infinitesimals to study the motion of the moon. The use of

infinitesimals to compute rates of change was developed significantly by Bhskara II

(1114-1185); indeed, it has been argued that many of the key notions of differential

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http://en.wikipedia.org/wiki/Bernhard_Riemannhttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Riemann_surfacehttp://en.wikipedia.org/wiki/Henri_Poincar%C3%A9http://en.wikipedia.org/wiki/Algebraic_topologyhttp://en.wikipedia.org/wiki/Dynamical_systemhttp://en.wikipedia.org/wiki/Complex_analysishttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Tangent_linehttp://en.wikipedia.org/wiki/Ancient_Greecehttp://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Apollonius_of_Pergahttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Infinitesimalhttp://en.wikipedia.org/wiki/Archimedes'_use_of_infinitesimalshttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Aryabhatahttp://en.wikipedia.org/wiki/Orbit_of_the_Moonhttp://en.wikipedia.org/wiki/Bh%C4%81skara_IIhttp://en.wikipedia.org/wiki/Bernhard_Riemannhttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Riemann_surfacehttp://en.wikipedia.org/wiki/Henri_Poincar%C3%A9http://en.wikipedia.org/wiki/Algebraic_topologyhttp://en.wikipedia.org/wiki/Dynamical_systemhttp://en.wikipedia.org/wiki/Complex_analysishttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Tangent_linehttp://en.wikipedia.org/wiki/Ancient_Greecehttp://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Apollonius_of_Pergahttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Infinitesimalhttp://en.wikipedia.org/wiki/Archimedes'_use_of_infinitesimalshttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Indian_mathematicshttp://en.wikipedia.org/wiki/Aryabhatahttp://en.wikipedia.org/wiki/Orbit_of_the_Moonhttp://en.wikipedia.org/wiki/Bh%C4%81skara_II


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calculus can be found in his work, such as "Rolle's theorem". The Persian mathematician,

Sharaf al-Dn al-Ts (1135-1213), was the first to discover the derivative of cubic

polynomials, an important result in differential calculus; his Treatise on Equations

developed concepts related to differential calculus, such as the derivative function and the

maxima and minima of curves, in order to solve cubic equations which may not have

positive solutions.

The modern development of calculus is usually credited to

Isaac Newton (1643 1727) and Gottfried Leibniz (1646 1716),

who provided independent and unified approaches to differentiation

and derivatives. The key insight, however, that earned them this credit,

was the fundamental theorem of calculus relating differentiation and integration: this

rendered obsolete most previous methods for computing areas and volumes, which had

not been significantly extended since the time of Ibn al-Haytham (Alhazen). For their

ideas on derivatives, both Newton and Leibniz built on significant earlier work bymathematicians such as Isaac Barrow (1630 1677), Ren Descartes (1596 1650),

Christiaan Huygens (1629 1695), Blaise Pascal (1623 1662) and John Wallis (1616

1703). Isaac Barrow is generallly given credit for the early development of the derivative.

Nevertheless, Newton and Leibniz remain key figures in the history of differentiation, not

least because Newton was the first to apply differentiation to theoretical physics, while

Leibniz systematically developed much of the notation still used today.

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http://en.wikipedia.org/wiki/Rolle's_theoremhttp://en.wikipedia.org/wiki/Islamic_mathematicshttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Maxima_and_minimahttp://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttp://en.wikipedia.org/wiki/Ibn_al-Haythamhttp://en.wikipedia.org/wiki/Isaac_Barrowhttp://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/John_Wallishttp://en.wikipedia.org/wiki/Theoretical_physicshttp://en.wikipedia.org/wiki/Rolle's_theoremhttp://en.wikipedia.org/wiki/Islamic_mathematicshttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Function_(mathematics)http://en.wikipedia.org/wiki/Maxima_and_minimahttp://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Gottfried_Leibnizhttp://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttp://en.wikipedia.org/wiki/Ibn_al-Haythamhttp://en.wikipedia.org/wiki/Isaac_Barrowhttp://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/John_Wallishttp://en.wikipedia.org/wiki/Theoretical_physics


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Since the 17th century many mathematicians have contributed to the theory of

differentiation. In the 19th century, calculus was put on a much more rigorous footing by

mathematicians such as Augustin Louis Cauchy (1789 1857), Bernhard Riemann

(1826 1866), and Karl Weierstrass (1815 1897). It was also during this period that the

differentiation was generalized to Euclidean space and the complex plane.

3. History of Progression

Geometric progressions have been found on Babylonian tablets dating back to

2100 BC. Arithmetic progressions were first found in the Ahmes Papyrus which is dated

at 1550 BC. The names for these notions, however, seem to have taken considerably

longer. In some cases there was no standard for how to refer to them (even the term

progression was not necessarily a standard).

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http://en.wikipedia.org/wiki/Augustin_Louis_Cauchyhttp://en.wikipedia.org/wiki/Augustin_Louis_Cauchyhttp://en.wikipedia.org/wiki/Bernhard_Riemannhttp://en.wikipedia.org/wiki/Karl_Weierstrasshttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/Augustin_Louis_Cauchyhttp://en.wikipedia.org/wiki/Bernhard_Riemannhttp://en.wikipedia.org/wiki/Karl_Weierstrasshttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Complex_plane


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The closest reasoning behind the names is that each term in a geometric

(arithmetic) sequence is the geometric (arithmetic) mean of it's successor and predessor.

The rationale behind the names of these means is a bit more clear: if we view the

quantitiesA andB as the lengths of the sides of a rectangle, then the geometric

mean is the length of the sides of a square having the same area as this rectangle.

This was viewed in those days as a verygeometric problem: finding the dimensions of a

square having the same area as a given figure (in this case, rectangle).

Although the arithmetic mean (A+B)/2 can also be interpreted geometrically (it is the

length of the sides of a square having the same perimeteras the rectangle), lengths were

viewed more as arithmetical concepts (because it's easy to handle lengths by ordinary

addition and subtraction, without having to think about two-dimensional concepts such as

area).

CONJECTURES

How mathematics is used in cake baking and cake decorating?

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Mathematics is widely used in baking and decorating without many people knowing

about it. Firstly, we can see the use of mathematical geometry where it is used in the

cake baking and cake decorating. In particular, mathematical geometry is used to

determine the dimension of the cake. In addition, it is used to assist in designing and

decorating cakes that come in many attractive shapes and designs and also inestimating the volume of cake to be produced.

On the other hand, the used of mathematics in cake baking and cake decorating

involves the use of calculus in particular differentiation. In cake baking and cake

decorating, differentiation is applied to determine the maximum and minimum

amount of ingredients for the cake baking. In addition, differentiation is used to

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estimate the minimum or maximum amount of cream needed for cake decorating

and also in estimating the minimum and maximum size of cake produced.

Apart from that, cake baking and cake decoration applied the usage of

mathematical progression where it is used to determine the total volume or weight of

multi-storey cakes with proportional dimensions. Not just that, progression is used to

estimate the total ingredients needed for cake baking and also in estimating the

amount of cream for decoration.

PROBLEM SOLVING

Question 1

To calculate the diameter of the baking tray to be used to fit the 5 kg :

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Given 1 kg cake has volume 3800cm, and h is 7.0 cm, so find d.(use = 3.142)

Volume of 5kg cake = rh

3800 5 = (3.142)(7.0)r

19000 = 21.994r

19000 =r

21.994

r = 863.872

r= 863.872

r= 29.392 cm

Notice that : d= 2r

d= 58.783 cm

Question 2

Given the inner dimensions of oven: 80cm length, 60cm width, 45cm height

a) To explore and tabulate different values of height, h cm, and the corresponding

values of diametersof the baking tray to be used, dcm.

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First, form the formula fordin terms ofh by using the above formula for volume of cake,

V = 19000, that is:

Volume of cake = rh

19000 = (3.142)(1.0)r

19000 = 3.142r

19000 =r

3.142

r = 6047.104

r= 6047.104

r= 77.763 cm

Notice that : d= 2r

d= 155.526 cm

Then, draw and complete table of 2 columns, 10 rows (example), as shown below: (use

that formula to find d, for every value of h)

Height, h (cm) Diameter, d(cm)

1.0 155.53

2.0 109.973.0 89.7934.0 77.7635.0 69.5536.0 63.4937.0 58.7838.0 54.9879.0 51.842

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10.0 49.181

b) Based on the values in table :

(i) To state and explain the range of heights that is not suitable for the cakes :

h < 7cm is NOT suitable, because the resulting diameter produced is too large to fit

into the oven. Furthermore, the cake would be too short and too wide, making it less

attractive.

(ii) The dimension that I think most suitable for the cake :

h = 8cm, d= 54.987cm, because it can fit into the oven, and the size is suitable for

easy handling.

c)

(i) To form an equation to represent the linear relation between h and d.

Use the same formula in Question 2(a), that is volume of cake = rh. But thin time

change rwith d/2. So, the formula is V = (d/2)h. The same process is also used,

that is, make dthe subject. This time, form an equation which is suitable and relevant

for the graph:

V = (d/2)h

19000 = (3.142)(d/2)h

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19000 d =

3.142h 4

6047.104 d

= h 4

24188.415 = d

h

155.53d=

h

d= 155.53h-

log10 d= log10 155.53h-

log10d= - log10h + log10 155.53

(the final equation for graph-drawing)

Another table is created, with one column is log10h

and the other is log10d

. Then, agraph oflog10dagainst log10h isplotted.

log10h (cm) log10d(cm)

0 2.1920.3010 2.0410.4771 1.9530.6021 1.8910.6989 1.8420.7781 1.803

0.8451 1.7690.9031 1.7400.9542 1.715

1.0 1.692

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(ii) Based on the graph :

(a) To determined the diameter of cake, dwhen height of cake, h is 10.5 cm :

when h = 10.5 cm, so log10 h is 1.021

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(b) To determined the height of cake, h when diameter of cake, dis 42 cm :

when d= 42 cm, so log10 dis 1.623

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Question 3

Best Bakery has been requested to decorate the cake with fresh cream. The

thickness of the cream is normally set to a uniform layer of about 1 cm.

(a) The amount of fresh cream required to decorated the cake using the dimension

that I have suuggested in 2(b)(ii) is estimated :

The dimension that I suggested in 2(b)(ii) is h = 8cm, d= 54.99cm.

Amount of fresh cream = Volume of fresh cream needed (area x height)

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Amount of fresh cream = Volume of cream at the top surface + Volume of cream at the

side surface

>>>Volume of cream at the top surface

= Area of top surface x Height of cream

= r x h (d/2) x h

= (3.142)(54.99/2) x 1

= 2375 cm

>>> Volume of cream at the side surface

= Area of side surface x Height of cream= (dx h) x Height of cream

= (3.142)(54.99)(8) x 1

= 1382.23 cm

Therefore, the amount of fresh cream = 2375 + 1382.23 = 3757.23 cm

(b) Three other shapes for cake, that will have the same height and volume as I

suggested in 2(b)(ii) is :

1. Rectangle

2. Triangle

3. Pentagon

Then, the amount of fresh cream to be used on each of the cakes is :

1 Rectangle-shaped base (cuboid)

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19000 = base area x heightbase area = 19000/8

length x width = 2375

By trial and improvement, 2375 = 50 x 47.5 (length = 50, width = 47.5, height = 8)

Therefore, volume of cream

= (Area of left side surface + Area of right side surface)(Height of cream) + (Area of

front side surface + Area of back side surface)(Height of cream) + Volume of top surface

= { [(8 x 50)+(8 x 50)](1) } + { [(8 x 47.5) + (8 x 47.5)](1) } + 2375 = 3935 cm

2 Triangle-shaped base

19000 = base area x height

base area = 2375

x length x width = 2375

length x width = 4750

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By trial and improvement, 4750 = 95 x 50 (length = 95, width = 50)

Slant length of triangle = (95 + 25)= 98.23

Therefore, amount of cream

= Area of rectangular front side surface(Height of cream) + 2(Area of slant rectangular

left/right side surface)(Height of cream) + Volume of top surface

= (50 x 8)(1) + 2(98.23 x 8)(1) + 2375 = 4346.68 cm

3 Pentagon-shaped base

19000 = base area x height

base area = 2375 = area of 5 similar isosceles triangles in a pentagon

therefore:

2375 = 5(length x width)

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475 = length x width

By 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) + volume of top surface

= 5(8 x 19) + 2375 = 3135 cm

(c) The shape that require the least amount of fresh cream to be used is determined :

Pentagon-shaped base cake, since it requires only 3135 cm of cream to be used.

Two different methods including Calculus is used t o find the dimension of a 5 kg

round cake that requires the minimum amount of fresh cream to decorate :

When there's minimum or maximum, well, there's differentiation and quadratic

functions. Use both to find the minimum height, h and its corresponding minimum

diameter, d.

Notice that

This question only gives just ONE info only: the mass of cake (which can change

into volume of cake, as shown)

So, change the 5 kg to 19000 cm

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Method 1: Differentiation

Use two equations for this method: the formula for volume of cake (as in Question2(a) ),

and the formula for amount (volume) of cream to be used for the round cake (as in

Question 3(a) ).

19000 = (3.142)rh (1)

V= (3.142)r + 2(3.142)rh (2)

From (1): h = 19000 /(3.142)r (3)

Sub. (3) into (2):

V= 3.142r + 2(3.142)(19000 / 3.142)r

V= 3.142r +38000 / r

V= 3.142r + 38000r-1

dV / dr= 2(3.142)r+ (-1)(38000)r-2

dV / dr= 6.284r38000r-2

dV / dr= 6.284r38000 / r2 -->> minimum value, therefore dV/ dr= 0

0 = 6.284r38000 / r2

38000 / r2 = 6.284r

38000 = 6.284r

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r = 6047.104

r= 6047.104

r= 18.22

Subtitude r= 18.22 into (1)

19000 = 3.142(18.22)h

19000 = 1043.04h

h = 18.22

Therefore, 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)rh (1)

f(r) = (3.142)r + 2(3.142)hr (2)

From (2):

f(r) = (3.142)(r + 2hr) -->> factorize (3.142)

Use the completing the square method

= (3.142)[ (r+ 2h/2) (2h/2) ]

= (3.142)[ (r+ h) h ]

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= (3.142)(r+ h) (3.142)h

a = (3.142) (positive indicates minimum value), minimum value = f(r) = (3.142)h,

corresponding value of x = r= (-h)

Sub. r= (-h) into (1):

19000 = (3.142)(-h)h

h = 6047.104h = 6047.104

h = 18.22

Sub. h = 18.22 into (1):

19000 = (3.142)r(18.22)

r = 331.894

r= 331.894

r= 18.22

therefore,

h= 18.22 cm,

d= 2

r= 2(18.22) = 36.44 cm

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I 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 handle easily.

FURTHER EXPLORATION

Given that height, h of each cake is constantly 6.0 cm. Then the radius of the first cake

which the largest cake is 31.0 cm. The radius of the second cake is less 10% less than the

radius of the first cake, the radius of the third cake is 10% less than the radius of the

second and so on.

From statement :

height, h of each cake = 6cm

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radius of largest cake = 31cm

radius of second cake = 90/100 x radius of first cake, 31.0 cm = 27.9 cm

radius of third cake = 90/100 x radius of second cake, 27.9 cm = 25.11 cm

etc.

So, conclusion is :

31, 27.9, 25.11, 22.599, , where geometric progression is used to determine the

radius of cake with first term is 31.0 and commonratio is 9/10 or 0.9.

a) To find the volume of the first, the second, the third and the fourth cakes :

Use the formula for volume V = (3.142)rh, with h = 6 to get the volume of cakes. The

values of rcan be obtained from the progression of radius of cakes given in previous

question.

Radius of 1st cake = 31, volume of 1st cake = (3.142)(31)(6) = 18116.772

Radius of 2nd cake = 27.9, vol. of 2nd cake = (3.142)(27.9)(6) = 14674.585

Radius of 3rd cake = 25.11, vol. of 3rd cake = (3.142)(25.11)(6) = 11886.414

Radius of 4th cake = 22.599, vol. of 4th cake = (3.142)(22.599)(6) = 9627.995

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Notice that :

9627.995428 11886.41411 = 0.81

11886.41411 14674.58532 = 0.81

14674.58532 18116.77200 = 0.81

Tn Tn-1= 0.81

Therefore, we can conclude that the volumes of the cakes form another geometric

progression with first term 18116.772 and common ratio0.81.

b) To calculate the maximum number of cakes that needs to be bake which the total

mass of all the cakes should not exceed 15 kg :

Notice that :

total mass < 15 kg, change to volume: total volume < 57000 cm

UseSn = a(1 - rn)1 r

whereSn = 57000, a = 18116.772 and r= 0.81 to find n:

Sn< 57000

a(1-rn) < 57000

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1r

18116.772(10.81n) < 57000

10.81

10830-0.81n < 1

18116.772

-0.81n< -0.4022

n log10 -0.81 < log10 -0.4022

n < log10 -0.4022-0.81

n < 4.322

n = 4

To verify the answer :

When n = 5,

S5 = a(1r5) 1r

S5 = 18116.772(10.815)

10.81

S5 = 62104.443

62104.443 > 57000

Sn > 57000, so n = 5 is not suitable.

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When n = 4,

S4 = a(1r4) 1r

S5 = 18116.772(10.814)

10.81

S5 = 54305.767

54305.767 < 57000

Sn < 57000, so n = 4 is suitable.

CONCLUSION

From all those activities I had done, I can conclude three main points :

Geometry can always be related to our surroundings. Geometry is very important

in everyday life. For examples, the knowledge of geometry is important to determine

suitable dimensions for the cake, to assist in designing and decorating cakes that

comes in many attractive shapes and designs, and to estimate volume of cake to

be produced.

Differentiation is used to determine the change of value of certain measurements,

depending on another related variable/measurement. In this project, differentiation is

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essential to determine minimum or maximum amount of ingredients for cake-

baking, to estimate minimum or maximum amount of cream needed for

decorating, and to estimate minimum or maximum size of cake produced.

Progression is very important in our environment and life. By using it, we are

able to determine total weight/volume of multi-storey cakes with proportional

dimensions, to estimate total ingredients needed for cake-baking, and to estimate

total amount of cream for decoration.

REFLECTION

This Additional Mathematics project had thought me various values and virtues. I

was very fortunate to have been given this project work because there are things I might

not be learning except by doing this project.

Firstly, I learn to be creative in finding solutions. This value is reflected when I'm

choosing the best shape for cake that is the most suitable. By being creative, I was able to

come out with various methods to find the minimum amount of fresh cream to decorate.

Similarly, thinking wise and creative helps you to solve problems in life. For instance, if I

always fail in my exams, there is possibility that my study methods are not right. Thus, I

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must create, figure out and change my methods of study, and this must be done by

thinking wise and creative. That's why it is essential in life.

Creativity allows us to create wonderful patterns and images

During the project work is completed, the various trials and challenges that I have

experienced. In the making of this project, I have spent countless hours doing this project.

Throughtout day and night

I sacrificed my precious time to have fun

From

Monday

Tuesday

Wednesday

Thursday

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Friday

And even on weekend

Fuhhhh

So tired

Sometimes too lazy to think...

Sometimes feel so boring...

Sometimes feel ... arghhh, all kinds of sense there ...

Of all the activities I have done, there are many values that I obtained during theexecution of this project. I can learn more in-depth Additional Mathematics. Also, I was

able to gain knowledge from my studies for this project work. This project work also

teach me to think out of the box.

School holidays this time are indeed strange for me. Normally I am not so hard

like this. But school holidays this time suddenly felt like to complete my school work,

odd as well, so why be so?. At first I just looked down on the work of this project. But

when other friends are busy talking about it on Facebook, all of a sudden I feel like want

to make it as well. So I am determined to complete it before the end of this holiday. So

Alhamdulillah, I managed to complete the project work of Additional Mathematics. This

may be because I was determined to complete it whatever is all.

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I make this project work using computers. While preparing this project work, I

heard nasyid songs. I'm really interested with the UNICs song, Satu Tekad. I am

interested in some lyrics of this song :

Satu Tekad (tekadku utk siap projek nie)

Berikan ku sebaris kata

Untuk ku susuri jalan gelita

Mentera keramat kata pujangga

Azimat yang bermaknaDi depan simpang bercabang tiga

Tak tahu mana satu arahnya

Kanan kiri manusia berdusta

Mungkir pada yang Esa

Mungkin ku bukan watak utama

Dalam pentas lakonan dunia

Bimbang juga ku turut sama

Pastinya aku yang binasa

Berikan aku pedoman (kawan2,ajarlah aq cmne nk wt nie)

Arah mana jalan kehidupan (aq x phm cmne nk wt la)

Untukku teruskan pengembaraan (aq nk dpt gak 10% markah utk trial nanti)

Tak rebah dan tersalah langkah (x nak kne mrh ngan cikgu Saadiah)

Tunjukkan aku kawan (cube bgi ckit solutionnye kwn2)

Simpang mana arah kejayaan (cmne nk siapkan projek ini)

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Timur, utara juga selatan

Atau arah matahari terbenam

"Bersama menyusuri jalan

Para rasul nabi junjungan

Menuju ke puncak gemilang terbilang

Iman di dada mengiring langkah"

As conclusion, I have learned many things from this project which are very useful

in my life. To be honest, this is why I love ADDITIONAL MATHEMATICS.

THE END

kerja kursus additional mathematics 2011

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