project work putrajaya 2013 : maximising a site

7/28/2019 Project Work Putrajaya 2013 : Maximising a Site

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Maximising A Site

Name : Ariff Imran Bin Mazlan

Class : 5 Seri Setia

Teacher : Sir Ng Seng Chew

School : Sekolah Menengah Kebangsaan Putrajaya Presint 14 (1)


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No. Title Page

1. Acknowledgement

2. Objectives

3. Introduction

4. Part 1

5. Part 26. Part 3

7. Further Exploration

8. Conclusion

9. Reflection


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First and foremost, I would like to thank my Additional

Mathematics teacher, Sir Ng Seng Chew as he gives us important

guidance and commitment during this project work. He has been a

very supportive figure throughout the whole project work.

I also would like to give thanks to all my friends for helping me

and always supporting me to help complete this project work. They

have done a great job at surveying various shops and sharing

information with other people including me. Without them thisproject would never have had its conclusion.

For their strong support, I would like to express my gratitude to

my beloved parents. Also for supplying the equipments and money

needed for the resources to complete this project. They have always

been by my side and I hope they will still be there in the future.

Last but not least, I would also like to thank all the nice

shopkeepers, staffs, and citizens for helping me collect the much

needed data and statistics for this. Not forgetting too all the other

people who were involved directly towards making this project a

reality.

I thank you all.


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The aims of carrying out this project work are to enable student to:

Use the language of mathematics to express mathematicalideas precisely.

Apply and adapt a variety of problem by solving strategies tosolve problem.

Use technology especially the ICT appropriately and effectively. To develop positive attitude towards mathematics. To promote effective mathematical. To improve thinking skills.


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An understanding of the attributes and relationships of

geometric objects can be applied in diverse contextsinterpreting a

schematic drawing, estimating the amount of wood needed to frame

a sloping roof, rendering computer graphics, or designing a sewingpattern for the most efficient use of material.

Although there are many types of geometry, school

mathematics is devoted primarily to plane Euclidean geometry,

studied both synthetically (without coordinates) and analytically

(with coordinates). Euclidean geometry is characterized most

importantly by the Parallel Postulate, that through a point not on a

given line there is exactly one parallel line. (Spherical geometry, in

contrast, has no parallel lines.)

During high school, students begin to formalize their geometry

experiences from elementary and middle school, using more precise

definitions and developing careful proofs. Later in college some

students develop Euclidean and other geometries carefully from a

small set of axioms.

The concepts of congruence, similarity, and symmetry can be

understood from the perspective of geometric transformation.

Fundamental are the rigid motions: translations, rotations,

reflections, and combinations of these, all of which are here assumed

to preserve distance and angles (and therefore shapes generally).

Reflections and rotations each explain a particular type of symmetry,

and the symmetries of an object offer insight into its attributesas


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when the reflective symmetry of an isosceles triangle assures that its

base angles are congruent.

In the approach taken here, two geometric figures are defined

to be congruent if there is a sequence of rigid motions that carriesone onto the other. This is the principle of superposition. For

triangles, congruence means the equality of all corresponding pairs of

sides and all corresponding pairs of angles. During the middle grades,

through experiences drawing triangles from given conditions,

students notice ways to specify enough measures in a triangle to

ensure that all triangles drawn with those measures are congruent.

Once these triangle congruence criteria (ASA, SAS, and SSS) areestablished using rigid motions, they can be used to prove theorems

about triangles, quadrilaterals, and other geometric figures.

Similarity transformations (rigid motions followed by dilations)

define similarity in the same way that rigid motions define

congruence, thereby formalizing the similarity ideas of "same shape"

and "scale factor" developed in the middle grades. These

transformations lead to the criterion for triangle similarity that twopairs of corresponding angles are congruent.

The definitions of sine, cosine, and tangent for acute angles are

founded on right triangles and similarity, and, with the Pythagorean

Theorem, are fundamental in many real-world and theoretical

situations. The Pythagorean Theorem is generalized to non-right

triangles by the Law of Cosines. Together, the Laws of Sines and

Cosines embody the triangle congruence criteria for the cases where

three pieces of information suffice to completely solve a triangle.

Furthermore, these laws yield two possible solutions in the

ambiguous case, illustrating that Side-Side-Angle is not a

congruencecriterion. Analytic geometry connects algebra and

geometry, resulting in powerful methods of analysis and problem

solving. Just as the number line associates numbers with locations in


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one dimension, a pair of perpendicular axes associates pairs of

numbers with locations in two dimensions. This correspondence

between numerical coordinates and geometric points allows

methods from algebra to be applied to geometry and vice versa. The

solution set of an equation becomes a geometric curve, making

visualization a tool for doing and understanding algebra. Geometric

shapes can be described by equations, making algebraic manipulation

into a tool for geometric understanding, odeling, and proof.

Geometric transformations of the graphs of equations correspond to

algebraic changes in their equations.

Dynamic geometry environments provide students withexperimental and modeling tools that allow them to investigate

geometric phenomena in much the same way as computer algebra

systems allow them to experiment with algebraic phenomena.


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The earliest recorded beginnings of geometry can be traced to

ancient Mesopotamia and Egypt in the 2nd millennium BC. 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 Egyptian Rhind Papyrus (20001800 BC) and Moscow

Papyrus (c. 1890 BC), the Babylonian clay tablets such as Plimpton

322 (1900 BC). For example, the Moscow Papyrus gives a formula for

calculating the volume of a truncated pyramid, or frustum. South of

Egypt the ancient Nubians established a system of geometry

including early versions of sun clocks. In the 7th century BC, the

Greek mathematician Thales of Miletus used geometry to solve

problems such as calculating the height of pyramids and the distance

of ships from the shore. He is credited with the first use of deductive

reasoning applied to geometry, by deriving four corollaries toThales'Theorem. Pythagoras established the Pythagorean School, which is

credited with the first proof of the Pythagorean theorem, though the

statement of the theorem has a long history Eudoxus (408c.355 BC)

developed the method of exhaustion, which allowed the calculation

of areas and volumes of curvilinear figures, as well as a theory of

ratios that avoided the problem ofincommensurable magnitudes,

which enabled subsequent geometers to make significant advances.
http://en.wikipedia.org/wiki/Mesopotamiahttp://en.wikipedia.org/wiki/Ancient_Egypthttp://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/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Babylonian_mathematicshttp://en.wikipedia.org/wiki/Plimpton_322http://en.wikipedia.org/wiki/Plimpton_322http://en.wikipedia.org/wiki/Frustumhttp://en.wikipedia.org/wiki/Nubiahttp://en.wikipedia.org/wiki/Greek_mathematicshttp://en.wikipedia.org/wiki/Thales_of_Miletushttp://en.wikipedia.org/wiki/Thales%27_Theoremhttp://en.wikipedia.org/wiki/Thales%27_Theoremhttp://en.wikipedia.org/wiki/Pythagoreanshttp://en.wikipedia.org/wiki/Pythagorean_theoremhttp://en.wikipedia.org/wiki/Eudoxus_of_Cnidushttp://en.wikipedia.org/wiki/Method_of_exhaustionhttp://en.wikipedia.org/wiki/Incommensurable_magnitudeshttp://en.wikipedia.org/wiki/Incommensurable_magnitudeshttp://en.wikipedia.org/wiki/Method_of_exhaustionhttp://en.wikipedia.org/wiki/Eudoxus_of_Cnidushttp://en.wikipedia.org/wiki/Pythagorean_theoremhttp://en.wikipedia.org/wiki/Pythagoreanshttp://en.wikipedia.org/wiki/Thales%27_Theoremhttp://en.wikipedia.org/wiki/Thales%27_Theoremhttp://en.wikipedia.org/wiki/Thales_of_Miletushttp://en.wikipedia.org/wiki/Greek_mathematicshttp://en.wikipedia.org/wiki/Nubiahttp://en.wikipedia.org/wiki/Frustumhttp://en.wikipedia.org/wiki/Plimpton_322http://en.wikipedia.org/wiki/Plimpton_322http://en.wikipedia.org/wiki/Babylonian_mathematicshttp://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Astronomyhttp://en.wikipedia.org/wiki/Constructionhttp://en.wikipedia.org/wiki/Surveyinghttp://en.wikipedia.org/wiki/Ancient_Egypthttp://en.wikipedia.org/wiki/Mesopotamia


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Around 300 BC, geometry was revolutionized by Euclid, whose

Elements, widely considered the most successful and influential

textbook of all time, introduced mathematical rigor through the

axiomatic method and is the earliest example of the format still usedin mathematics today, that of definition, axiom, theorem, and proof.

Although most of the contents of the Elements were already known,

Euclid arranged them into a single, coherent logical framework. The

Elements was known to all educated people in the West until the

middle of the 20th century and its contents are still taught in

geometry classes today.Archimedes (c.287212 BC) ofSyracuse used

the method of exhaustion to calculate the area under the arc of

a parabola with the summation of an infinite series, and gave

remarkably accurate approximations ofPi. He also studied

the spiral bearing his name and obtained formulas for

the volumes ofsurfaces of revolution.

In the Middle Ages, mathematics in medieval Islam contributed

to the development of geometry, especially algebraic geometry

andgeometric algebra. Al-Mahani 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 arithmetic operations applied to ratios of geometrical

quantities, and contributed to the development ofanalytic

geometry. Omar Khayym (10481131) found geometric solutions

to cubic equations. The theorems ofIbn al-Haytham(Alhazen), Omar

Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including

the Lambert quadrilateral and Saccheri quadrilateral, were early

results in hyperbolic geometry, and along with their alternative

postulates, such as Playfair's axiom, these works had a considerable

influence on the development of non-Euclidean geometry among
http://en.wikipedia.org/wiki/Euclid%27s_Elementshttp://en.wikipedia.org/wiki/Mathematical_rigorhttp://en.wikipedia.org/wiki/Axiomatic_methodhttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Syracuse,_Italyhttp://en.wikipedia.org/wiki/Method_of_exhaustionhttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Parabolahttp://en.wikipedia.org/wiki/Series_(mathematics)http://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Archimedes_spiralhttp://en.wikipedia.org/wiki/Volumehttp://en.wikipedia.org/wiki/Surface_of_revolutionhttp://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/Th%C4%81bit_ibn_Qurrahttp://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/Analytic_geometryhttp://en.wikipedia.org/wiki/Omar_Khayy%C3%A1mhttp://en.wikipedia.org/wiki/Cubic_equationhttp://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/Saccheri_quadrilateralhttp://en.wikipedia.org/wiki/Hyperbolic_geometryhttp://en.wikipedia.org/wiki/Playfair%27s_axiomhttp://en.wikipedia.org/wiki/Playfair%27s_axiomhttp://en.wikipedia.org/wiki/Hyperbolic_geometryhttp://en.wikipedia.org/wiki/Saccheri_quadrilateralhttp://en.wikipedia.org/wiki/Lambert_quadrilateralhttp://en.wikipedia.org/wiki/Quadrilateralhttp://en.wikipedia.org/wiki/Nasir_al-Din_al-Tusihttp://en.wikipedia.org/wiki/Ibn_al-Haythamhttp://en.wikipedia.org/wiki/Cubic_equationhttp://en.wikipedia.org/wiki/Omar_Khayy%C3%A1mhttp://en.wikipedia.org/wiki/Analytic_geometryhttp://en.wikipedia.org/wiki/Analytic_geometryhttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Latinhttp://en.wikipedia.org/wiki/Th%C4%81bit_ibn_Qurrahttp://en.wikipedia.org/wiki/Al-Mahanihttp://en.wikipedia.org/wiki/Geometric_algebrahttp://en.wikipedia.org/wiki/Algebraic_geometryhttp://en.wikipedia.org/wiki/Mathematics_in_medieval_Islamhttp://en.wikipedia.org/wiki/Middle_Ageshttp://en.wikipedia.org/wiki/Surface_of_revolutionhttp://en.wikipedia.org/wiki/Volumehttp://en.wikipedia.org/wiki/Archimedes_spiralhttp://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Series_(mathematics)http://en.wikipedia.org/wiki/Parabolahttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Method_of_exhaustionhttp://en.wikipedia.org/wiki/Syracuse,_Italyhttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Axiomatic_methodhttp://en.wikipedia.org/wiki/Mathematical_rigorhttp://en.wikipedia.org/wiki/Euclid%27s_Elements


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later European geometers, including Witelo, Gersonides, Alfonso,

John Wallis, and Giovanni Girolamo Saccheri.

In the early 17th century, there were two important

developments in geometry. The first was the creation of analytic

geometry, or geometry withcoordinates and equations, by Ren

Descartes (15961650) 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 a geometrywithout measurement or parallel lines, just the study of how points

are related to each other.

Two developments in geometry in the 19th century changed

the way it had been studied previously. These were the discovery

ofnon-Euclidean geometries by Nikolai Ivanovich Lobachevsky

(17921856), Jnos Bolyai (18021860) and Carl Friedrich Gauss

(17771855) and of the formulation ofsymmetry as the central

consideration in the Erlangen Programme ofFelix Klein (which

generalized the Euclidean and non-Euclidean geometries). Two of the

master geometers of the time were Bernhard Riemann (18261866),

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.
http://en.wikipedia.org/wiki/Witelohttp://en.wikipedia.org/wiki/Gersonideshttp://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/Equationhttp://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttp://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_Kleinhttp://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/Complex_analysishttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Complex_analysishttp://en.wikipedia.org/wiki/Complex_analysishttp://en.wikipedia.org/wiki/Dynamical_systemhttp://en.wikipedia.org/wiki/Algebraic_topologyhttp://en.wikipedia.org/wiki/Henri_Poincar%C3%A9http://en.wikipedia.org/wiki/Riemann_surfacehttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Bernhard_Riemannhttp://en.wikipedia.org/wiki/Felix_Kleinhttp://en.wikipedia.org/wiki/Erlangen_Programmehttp://en.wikipedia.org/wiki/Symmetryhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/J%C3%A1nos_Bolyaihttp://en.wikipedia.org/wiki/Nikolai_Ivanovich_Lobachevskyhttp://en.wikipedia.org/wiki/Non-Euclidean_geometryhttp://en.wikipedia.org/wiki/Girard_Desargueshttp://en.wikipedia.org/wiki/Projective_geometryhttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttp://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttp://en.wikipedia.org/wiki/Equationhttp://en.wikipedia.org/wiki/Coordinate_systemhttp://en.wikipedia.org/wiki/Giovanni_Girolamo_Saccherihttp://en.wikipedia.org/wiki/John_Wallishttp://en.wikipedia.org/wiki/Alfonsohttp://en.wikipedia.org/wiki/Gersonideshttp://en.wikipedia.org/wiki/Witelo


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Woman teaching geometry. Illustration at the beginning of a

medieval translation ofEuclid's Elements,

A European and an Arab practicing geometry in the 15th century.
http://en.wikipedia.org/wiki/Euclid%27s_Elementshttp://en.wikipedia.org/wiki/Europehttp://en.wikipedia.org/wiki/Arabhttp://en.wikipedia.org/wiki/File:Woman_teaching_geometry.jpghttp://en.wikipedia.org/wiki/File:Westerner_and_Arab_practicing_geometry_15th_century_manuscript.jpghttp://en.wikipedia.org/wiki/Arabhttp://en.wikipedia.org/wiki/Europehttp://en.wikipedia.org/wiki/File:Westerner_and_Arab_practicing_geometry_15th_century_manuscript.jpghttp://en.wikipedia.org/wiki/Euclid%27s_Elements


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Geometry (AncientGreek:; geo- "earth", -metron

"measurement") is a branch ofmathematics concerned with

questions of shape, size, relative position of figures, and the

properties of space. A mathematician who works in the field of

geometry is called a geometer. Geometry arose independently in a

number of early cultures as a body of practical knowledge

concerning lengths, areas, and volumes, with elements of a formal

mathematical science emerging in the West as early as Thales (6th

Century BC). By the 3rd century BC geometry was put into

an axiomatic formby Euclid, whose treatmentEuclidean

geometryset a standard for many centuries to

follow. Archimedes developed ingenious techniques for calculating

areas and volumes, in many ways anticipating modern integral

calculus. The field ofastronomy, especially mapping the positions of

thestars and planets on the celestial sphere and describing therelationship between movements of celestial bodies, served as an

important source of geometric problems during the next one and a

half millennia. Both geometry and astronomy were considered in the

classical world to be part of theQuadrivium, a subset of the

seven liberal arts considered essential for a free citizen to master.
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The introduction ofcoordinates by Ren Descartes and the

concurrent developments ofalgebra marked a new stage for

geometry, since geometric figures, such as plane curves, could now

be represented analytically, i.e., with functions and equations. Thisplayed a key role in the emergence ofinfinitesimal calculus in the

17th century. Furthermore, the theory ofperspective showed that

there is more to geometry than just the metric properties of figures:

perspective is the origin ofprojective geometry. The subject of

geometry was further enriched by the study of intrinsic structure of

geometric objects that originated with Euler and Gauss and led to the

creation oftopology and differential geometry.

In Euclid's time there was no clear distinction between physical

space and geometrical space. Since the 19th-century discovery

ofnon-Euclidean geometry, the concept ofspace has undergone a

radical transformation, and the question arose: which geometrical

space best fits physical space? With the rise of formal mathematics in

the 20th century, also 'space' (and 'point', 'line', 'plane') lost its

intuitive contents, so today we have to distinguish between physical

space, geometrical spaces (in which 'space', 'point' etc. still have their

intuitive meaning) and abstract spaces. Contemporary geometry

considers manifolds, spaces that are considerably more abstract than

the familiar Euclidean space, which they only approximately

resemble at small scales. These spaces may be endowed with

additional structure, allowing one to speak about length. Modern

geometry has multiple strong bonds with physics, exemplified by the

ties between pseudo-Riemannian geometry and general relativity.

One of the youngest physical theories, string theory, is also very

geometric in flavour.
http://en.wikipedia.org/wiki/Coordinateshttp://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Plane_curvehttp://en.wikipedia.org/wiki/Analytic_geometryhttp://en.wikipedia.org/wiki/Infinitesimal_calculushttp://en.wikipedia.org/wiki/Perspective_(graphical)http://en.wikipedia.org/wiki/Projective_geometryhttp://en.wikipedia.org/wiki/Eulerhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Differential_geometryhttp://en.wikipedia.org/wiki/Non-Euclidean_geometryhttp://en.wikipedia.org/wiki/Spacehttp://en.wikipedia.org/wiki/Manifoldhttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Pseudo-Riemannianhttp://en.wikipedia.org/wiki/General_relativityhttp://en.wikipedia.org/wiki/String_theoryhttp://en.wikipedia.org/wiki/String_theoryhttp://en.wikipedia.org/wiki/General_relativityhttp://en.wikipedia.org/wiki/Pseudo-Riemannianhttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Manifoldhttp://en.wikipedia.org/wiki/Spacehttp://en.wikipedia.org/wiki/Non-Euclidean_geometryhttp://en.wikipedia.org/wiki/Differential_geometryhttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Eulerhttp://en.wikipedia.org/wiki/Projective_geometryhttp://en.wikipedia.org/wiki/Perspective_(graphical)http://en.wikipedia.org/wiki/Infinitesimal_calculushttp://en.wikipedia.org/wiki/Analytic_geometryhttp://en.wikipedia.org/wiki/Plane_curvehttp://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttp://en.wikipedia.org/wiki/Coordinates


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While the visual nature of geometry makes it initially more

accessible than other parts of mathematics, such as algebra

or number theory, geometric language is also used in contexts far

removed from its traditional, Euclidean provenance (for example,in fractal geometry and algebraic geometry)
http://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Fractal_geometryhttp://en.wikipedia.org/wiki/Algebraic_geometryhttp://en.wikipedia.org/wiki/Algebraic_geometryhttp://en.wikipedia.org/wiki/Fractal_geometryhttp://en.wikipedia.org/wiki/Number_theory


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SQUARE

All sides of a square are congruent. All angles are right

angles. The sum of the interior angles of a square will always

be 360 degrees. A square is a rectangle, while a rectangle is

not a square. A square is under the category of quadrilaterals

along with the rectangle, trapezoid, kite, and rhombus.


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RECTANGLE

The both pairs of opposite sides of rectangles are parallel.

Both pairs of opposite sides of rectangles are congruent.

All angles of rectangle are 90 degrees. Both pairs ofopposite angles are congruent.The diagonals bisect each

other.


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CIRCLE

A circle is a shape where every point along the edge is

equidistant from a point at the center. The distance from the

center to the edge is called the radius. The distance across

the widest point of a circle is called the diameter, and it is

always equal to the radius x 2. The distance around the circle

is called the circumference, and is equal to the diameter x 2 x

pi (3.14 and so on). The area of the circle is equal to the

radius squared (multiplied by itself) x pi.


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TRIANGLE

Any 3-sided polygon whose internal angles equal to 180

degrees. The length of all side of triangle are equal. Got 3

side and vertices.


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PENTAGON

Number of sides of a pentagon is 5 side. The number of

vertices are also 5. The interior angle of pentagon is

108.Meanwhile, the exterior angle of pentagon is 72. The

number of side of a pentagon are equal to the number of

vertices. The Exterior angle of a pentagon is multiplied by

the number of side is 360.


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TRAPEZOID

The only real characteristic of a trapezoid is that one pair of

opposite sides is parallel. For an isosceles trapezoid, in

addition to one pair of opposite sides being parallel, the legs

are congruent, each pair of base angles is congruent, and thediagonals are congruent.


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Question (a) :

The school has chosen a rectangular-shape site for the

landscape. By using appropriate variables, express the area of

the park site.

Park Site

VARIABLES :

1.Length of the brick2.Width of the brick3.Area of the park site4.Width of the park site5.Length of the park site6.Shape of the park site


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AREA :

Perimeter of the park site,

Area of the park site :


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Question (b) :

Use at least two methods to determine the maximum area of

the site. Further, find the length of the base area of the site in

order to determine maximum area.

I. Differentiation :

When Thus, the maximum area of the site is 3.0625


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II. Graphical MethodPlot graph against . Based on the table of values ofand

Length of site area, x (m) Area of site area, A (m)

1.70 3.0600

1.71 3.06091.72 3.0616

1.73 3.0621

1.74 3.0624

1.75 3.0625

1.76 3.0624

1.77 3.0621

1.78 3.0616

1.79 3.0609

1.80 3.0600


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when Thus, the maximum area of the site is .

3.0595

3.06

3.0605

3.061

3.0615

3.062

3.0625

3.063

1.68 1.7 1.72 1.74 1.76 1.78 1.8 1.82

A(m)


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Question (c) :

Determine the minimum number of brick required. If the cost

of purchasing a piece of brick is RM7.00, calculate the

minimum cost to build the area of the site. (assume that the

brick have equal length)

The minimum number of bricks required is :

Solution:

The perimeter of the park site = 7.0m

The length of 1 bricks = 0.2m

The minimum number of bricks required is =

= 35 pieces

The cost of 1 bricks required is = RM7.00

The minimum cost making the site area,

= The minimum number of bricks required x The cost of 1

bricks

= 35 pieces x RM7.00

= RM245.00


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You plan to transform the Floria Park site from rectangularto trapezoidal shape. The park is decorated with twocircular ponds of different sizes as shown in the figures

below. The remaining area of the garden planted with

flowers.

(a)The area of the flower site is and the radius of the fishpond is . Express in term of and .


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6m

Big Pond

Luas Trapezium Luas Kolam Besar Luas Kolam Kecil

Small Pond


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(b)Find the diameter of the large fish pond if the site is

120. (Used (Give your answer correct to 4significant figures).

Therefore, The diameter of large fish pond is 4.566.


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(c)Fish pond construction cost is higher than the cost of

building a flower site. Use two methods to determine thearea of flower site in order to minimize the construction cost

(Use ).

i. Differentiation

)

When


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When x is maximum

The cost of constructing the garden is minimum when thearea of flower site is .


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ii. Tabulation

In order that the cost of the fish ponds to be minimum, find

the maximum area of flower site.

0.5 117.580

1 122.292

1.5 123.863

2 122.292

2.5 117.580

3 109.7263.5 98.731

4 84.593

4.5 67.314

5 46.894

The cost of constructing the garden is minimum when thearea of flower site is .


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If the area of the garden remains the same, build thegarden with other form of shape to obtain minimum

perimeter. Explain your reason.

Compare 3 different shape :

A Square :

Perimeter


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A Circle

Perimeter


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A Semi-Circle

Perimeter


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As the conclusion to this project work, after doing research,

answering questions, drawing graphs and some problem

solving, I notice that geometrical shape are close with us in

our daily life. That is because all the things or anything are

made up using geometrical shape and all the geometrical

shape got its own name. Trapezium is one of it, The shape

can be use for design or pattern for anything such as

landscape design and decoration like what I did in this project

work. This project work also help student to improve their

mathematics calculation such as differentiation, quadratic

equation, geometry, and other thing that required to solve

the question given, and also tell the student that

mathematics is usefull, it can be used in our daily life. Actually

mathematics is everywhere and may on this green planet, we

may not recognized it because it doesnt look like the

mathematics we did in school. Mathematics in the world

around us sometimes seems invisible. But mathematics is

present in our world all the time in the workplace, in our

homes and in life generally.I

ADDMATHS!


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Not only this project encourages the student to think

critically to identify and solve problems. It is also encourage

student to gather information using the technologies such as

the internet, improve thinking skills and promote effective

mathematical communication.After spending countless

hours, days and night to finish this project and also sacrificing

my facebook and skyping time in this mid year holiday, there

are several things that I can say...

If people do not believe that additionalmathematics is simple, it is only because theydo not realize how complicated life is.

project work putrajaya 2013 : maximising a site

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