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    SEKOLAH MENENGAH KEBANGSAAN TUNKU ABDUL

    RAHMAN NIBONG TEBAL

    Jalan Bukit Panchor,

    14300 Nibong Tebal.

    Seberang Perai Selatan,

    Pulau Pinang.

    PROJECT WORK FOR ADDITIONAL MATHEMATICS 2010

    PROJECT WORK 1

    SCHOOL : SEKOLAH MENENGAH KEBANGSAAN TUNKU ABDUL

    RAHMAN

    CLASS : 5 USM

    TEACHER : PN. ZUWARIYAH BINTI ABDUL WAHAB

    NAME : LIM YEE THING

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    I/C : 930619 07 5358

    TABLE OF CONTENT

    No Title Page Number1 Introduction 32 Acknowledgement 43 History 54 Objective 65 Task specification 7-126 Further exploration 13-177 Conclusion 188 Reflection 19

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    INTRODUCTION

    Math is the real world, its everywhere.

    I get this sentence from the famous American Series NUMB3RS.

    Mathematics is the study of quantity, structure, space and change.

    Mathematicians seek out patterns, formulate new conjectures, and

    establish truth by rigorous deduction from appropriately chosen

    axioms and definitions. We use mathematics to solve simple

    mathematic equation and compare the price of product in our daily

    life. In fact, mathematics knowledge has been applied in various

    fields such as astronomy, business, computing, architecture and the

    list goes on. In the NUMB3RS, famous American Series, it shows that

    mathematics can be used to predict criminal activities.

    Furthermore, mathematics is also being used in the construction

    of buildings. With mathematics, we can determine the height of

    building, the stability of the building against wind resistance or

    earthquake, and the shape of the building which is the most ideal

    and unique. Besides, we can calculate the cost needed to construct

    a building. Mathematics also helps us to save up cost of

    construction by determining the shape of the building which built.

    An architect must have this knowledge in order to predict and build

    the building exactly the cost that is required to spend.

    We can eventually do many things that are beyond our

    imagination if we further explore the great mystery of mathematics.

    Therefore I am very sure that I will certainly attain and learn

    something which benefits me in daily life.

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    ACKNOWLEDGEMENT

    First of all, I would like to thank sincerely to my parents for

    providing me everything that is required to complete this project.For instance, they give me money to buy anything related to this

    project work, their advice, and facilities such as computers, books,

    and reference material. My parents greatly supported and

    encouraged me in order to build my confidence to do this task so

    that I will not procrastinate in doing it.

    Furthermore, I would like to say thank you to my teacher, Pn.

    Zuwariyah, for guiding me throughout this project. She taught us

    patiently to do the project, without any complaint. Even when I met

    some hardship and difficulties, she taught me patiently until I got

    what she mean and what must I do. She tried and did not give up

    teaching me until I finished the project successfully.

    In addition, my friends also helped me a lot while doing this task.

    Although this project is individual project, we discussed and

    cooperated to get more information about this project in order to

    understand more and make the project more perfect. Our

    cooperation does really help each of us a lot in completing this task.

    Lastly, those who helped me in this project indirectly, I would

    really want to appreciate all of you. Thank you.

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    HISTORY

    The product rule and chain rule, the notion of higher

    derivatives, Taylor series, and analytical functions were introducedby Isaac Newton in an idiosyncratic notation which he used to solve

    problems of mathematical physics. In his publications, Newton

    rephrased his ideas to suit the mathematical idiom of the time,

    replacing calculations with infinitesimals by equivalent geometrical

    arguments which were considered beyond reproach. He used the

    methods of calculus to solve the problem of planetary motion, the

    shape of the surface of a rotating fluid, the oblate ness of the earth,

    the motion of a weight sliding on a cycloid, and many other

    problems discussed in his Principia Mathematics. In other work, he

    developed series expansions for functions, including fractional and

    irrational powers, and it was clear that he understood the principles

    of the Taylor series.

    These ideas were systematized into a true calculus of

    infinitesimals by Gottfried Wilhelm Leibniz, who was originally

    accused of plagiarism by Newton. He is now regarded as an

    independent inventor of and contributor to calculus. His contribution

    was to provide a clear set of rules for manipulating infinitesimal

    quantities, allowing the computation of second and higher

    derivatives, and providing the product rule and chain rule, in their

    differential and integral forms. Unlike Newton, Leibniz provided were

    the laws of differentiation and integration, second and higher

    derivatives, and the notion of an approximating polynomial series.

    By Newtons time, the fundamental theorem of calculus was known.

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    OBJECTIVE

    The aims of carrying out this project work are:

    1. To provide learning environment that stimulates and enhances

    effective learning.

    2. To apply and adapt a variety of problem-solving strategies to

    solve problems.

    3. To promote effective mathematical communication.

    4. To use the language of mathematics to express mathematical

    ideas precisely.

    5. To develop positive attitude towards mathematics.

    6. To develop mathematical knowledge through problem solving

    in a way that increases students interest and confidence.

    7. To improve thinking skills.

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    TASK SPECIFICATION

    The diagram below shows the gate of an art gallery. A concrete

    structure is built at the upper part of the gate and the words ARTGALLERY is written on it. The top of the concrete structure is flat

    whereas the bottom is parabolic in shape. The concrete structure is

    supported by two vertical pillars at both ends.

    The distance between the two pillars is 4 metres and the height of

    the pillar is 5 metres. The height of the concrete structure is 1

    metre. The shortest distance from point A of the concrete structure

    to point B, that is the highest point on the parabolic shape, is 0.5

    metres.

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    PROBLEM SOLVING

    (a)The parabolic shape of the concrete structure can be representedby various functions depending on the point of reference. Based

    on different points of reference, obtain at least three different

    functions which can be used to represent the curve of this

    concrete structure.

    For function 1:

    Maximum point (0,4.5) and pass through point (2,4)

    b=0, c=4.5

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    Substitute (2,4) into (1)

    For function 2:

    Since (0, ) is the maximum point of the equation,

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    Substitute point (2, 0) into equation 2,

    Function 2:

    For function 3:

    Maximum point (2,4.5) and pass through point (0,4)

    b=2, c=4.5

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    Substitute (0,4) into (3)

    (b)The front surface of this concrete structure will be painted before

    the words ART GALEERY is written on it. Find the area to be

    painted.

    Using Function 1,

    Area of shaded region

    = Area of painted region

    = Area of rectangle Area of curve from origin

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    = (4 x 1) -

    = 4 -

    = 4 -

    = 4 -

    = 4 -

    = 4 - 1

    = 2

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    FURTHER EXPLORATION

    (a)You are given four different shapes of concrete structure as

    shown in the diagrams below. All the structures have the same

    thickness of 40 cm and are symmetrical.

    STRUCTURE 1 STRUCTURE 2

    STRUCTURE 3 STRUCTURE 4

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    (i) Given that the cost to construct 1 cubic metre of concrete is

    RM840.00, determine which structure will cost the minimum to

    construct.

    Structure 1Ignoring the pillars,

    Area = 2

    Volume = 2 x 0.4

    =1

    Cost needed =1 x RM840

    =RM896.00

    Structure 2

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    Ignoring the pillars,

    Area = 4(0.5)+ 2( )(2)(0.5)

    = 3

    Volume = 3 x 0.4

    =1.2

    Cost needed =1.2 x RM840

    =RM1008.00

    Structure 3Ignoring the pillars,

    Area = 4(0.5)+ 1.5(0.5)

    = 2.75

    Volume = 2.75 x 0.4

    =1.1

    Cost needed =1.1 x RM840

    =RM924.00

    Structure 4Ignoring the pillars,

    Area =