SEKOLAH MENENGAH KEBANGSAAN

(Your School Name)

ADDITIONAL MATHEMATICS

PROJECT WORK 1/2010

Title : The Gate of WisDom

Name :

I.C. no. :

Class :

Teacher :

Content

Title PagePreface 1

Appreciation 2

Objective 3

Introduction 4 - 13

Part a(і) 15

Part a(іі) 16

Part a(ііі) 17

Part b 18

Further Exploration 19 - 26

Conclusion 27

Reflection 28

Reference 29

PREFACE

I have done many researches throughout the internet and discussing with a friend who have helped me a lot in completing this project. Through the completion of this project, I have learned many skills and techniques. This project really helps me to understand more about the uses of integration in our daily life.

This project also helped expose the techniques of application of additional mathematics in real life situations.

While I was conducting this project work, I have gained consciousness in many other things in my life. Completing this project work in a team had gave me a chance to know each other and broaden my view on Mathematics. I’m now know the correct way to apply mathematic knowledge in my daily life to solve a lot of problems.

Besides that, I also have learnt to accept other ideas from different people to make out all the possible results. Then, I know which is the best after those comparison made.

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Appreciation

First of all, I want to express my utmost gratitude to God, for giving me the strength and health to do this project.

I would like to thank my parents for providing me everything, such as money, to buy anything that related to this project work

and their advise and support to me, which is the most needed to complete this project work.

Not forgotten my add maths teacher, Miss. Chung for guiding me and my friends throughout this project. We had some difficulties in doing this project work, but she taught us patiently until we knew what to do. She tried and tried to teach us until we understand what we supposed to do with this project work.

Last but not least, my friends who were doing this project with me and sharing our ideas.

They were helpful that when we combined and discussed together, we had this project work done.

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Objective

The aims of carrying out this project work are :

to apply ad adapt a variety of problem-solving strategies to solve problems ;

to improve thinking skills ;

to promote effective mathematical communication ;

to develop mathematical knowledge through problem solving in a way that increase student’s interest and confidence ;

to use the language of mathematics to express mathematical ideas precisely ;

to provide learning environment that stimulates and enhances effective learning ;

to develop positive attitude towards mathematics ;

to learn the way to reply the formulas of mathematics in our daily life accurately.

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Arch

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The arch is an incredible architectural discovery, dating back to ancient times but still in wide use today, as, up until the 19th century, it was the only known method for roofing a building without the use of beams. It comes in many shapes-semicircular (Roman), segmental (less than half a circle), or pointed (Gothic). The arch developed from the post and lintel or possibly the corbel, which is similar in shape and principle to the arch. Efforts to build corbelled roofs with smaller until and less weight could have eventually led to the discovery of the arch.

Arches are made of wedge-shaped blocks, called voussoirs, set with their narrow side toward the opening so that they lock together. The topmost voussoir is called the keystone. Once locked into place, the arch cannot collapse under any amount of weight and the only danger is of the voussiors crumbling under the pressure. To keep this from happening, most arches require support from other arches, walls, or buttresses.

The arch has been found in many different cultures, as early as Mesopotamia. The Egyptians used it in tombs and vaults but never for monumental architecture, such as temples. They apparently thought it unsuited to this purpose. The Greek also used the arch solely for practical constructions, but many of the principles they developed were later exploited by the Romans.

Overall, it was not until the time of the Etruscans that the arch was used in any kind of monument. The best example of this is the Porta Augusta, where the arch is combined with Greek architectural ideas. The Romans borrowed this combination and used it over and over again, but its invention belongs solely to the Etruscans.

The Romans took many great strides in the development of the arch. While they borrowed many techniques from earlier races, the Romans invented the idea of setting an arch on top of two tall pedestals to span a walkway such as a public highway. The outer wall of the Colosseum appears composed almost completely of arches. Here we see examples of the barrel vault and the more complicated groined vault, both developed by the Romans from the basic arch. The Romans also used arches for common purposes, such as in the building of bridges and aqueducts.

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Uses of Quadratic Equation

Quadratic equations are used in many areas of science and engineering. The path of a projectile (e.g. a cannon ball) is (almost) parabolic, and we use a quadratic equation to find out where the projectile is going to hit. Also, parabolic antennas are another application.

Real mathematics is about modelling situations that occur naturally, and using the model to understand what is happening, or maybe to predict what will happen in future.

The quadratic equation is often used in modelling because it is a beautifully simple curve.

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History of the Quadratic Equation

Throughout the years, the history of mathematics has taken its fair share of changes. It has stretched across the world from the Far East, migrating into the Western Hemisphere.

One of the most fundamental and key principles of mathematics has been the quadratic formula. Having been used in several different cultures, the formula has been part of the base of mathematics theory.

The general equation has been derived from many different sources, most commonly: ax2 + bx + c = 0, with x being the variable and a, b, and c its respective constant terms.

Though this is how modern mathematics perceives the equation, different symbols and notations have been used to represent the formula.

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Why we have to study quadratic equation?

See the diagrams and photos below. They will enlighten you

The communication dish is parabolic in shape. Parabolic is the equivalent to quadratic mathematically. Engineers need to understand quadratic equation to design this beautiful profile

This wok is designed using quadratic expression. With this, food can be fried to our liking!

Without quadratic equation, who knows how a wok would look like.

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Here you see that eye-glass lens are constructed with curves matching that of the quadratic equation.

Light is thus controlled to give good image to our eyes.

Quadratic equations to the rescue, right?

Other examples are:

1) Distance travelled given by the quadratic equation s = ut + (1/2) a t2

2) Electrical characteristics of a MOSFET (Transistor device)i = k [(Vg - Vt)VD - (1/2)Vd2]

So now do you still wonder why you study quadratic equations?

Maths do have a purpose in our daily life. Rest assure that you are studying maths for a good cause.

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Question A)

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 'ART GALLERY' 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.

(a) The parabolic shape of the concrete structure can be represented by 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.

In this part of question, I will use the principles of quadratic functions with three different maximum point to obtain three different functions.

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Part a(іі)

(іі) As a president of the Art club, you are given the opportunity to decide on the shape of the gate to be constructed. Which shape would you choose? Explain and elaborate on your reasons for choosing the shape.

As the president of the Arts Club, I will choose structure 1. This is because the cost needed is worthwhile. We can construct a rigid structure by maximizing our structure's performance by using only the smallest amount of materials needed.

Secondly, this structure is more stable compared to the others as the pressure exerted on it is distributed evenly to all parts of the structure because of its shape. In this way, it can withstand higher pressure. Therefore, it lasts longer compared to other structures and needed less frequent maintenance.

Furthermore, based on survey conducted, majority people felt uneasy when they saw sharp edges on most objects. Since from the beginning of time, a lot of people considered circular objects are the most perfect objects. Beside that, structure 1 also give better support to the gate because of its parabolic shape. In addition, the parabolic shape matches the standard of society nowadays.

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b (іі) From the values of the area to be painted from Table 1. I notice that:

There is a pattern in the area to be painted

The area to be painted decreases as the k increases 0.25m and form a series of numbers :

3, 2.9375, 2.875, 2.8125, 2.75, 2.6875, 2.625, 2.5625, 2.5

We can see that the difference between each term and the next term is the same.

2.9375 – 3 = -0.0625

2.875 – 2.9375 = - 0.0625

2.8125 – 2.875 = - 0.0625

2.75 – 2.8125 = - 0.0625

2.6827 – 2.75 = - 0.0625

2.625 – 2.6827 = - 0.0625

2.5625 – 2.625 = - 0.0625

2.5 – 2.5625 = - 0.0625

We can deduce that the area to be painted also forms an arithmetic progression with a common difference of -0.0625

In conclusion, when k increases 0.25m, the area to be painted decreased by 0.0625m2.

(C) Express the area of the concrete structure to be painted in terms of k. Find the area of k approaches the value of 4 and predict the shape of the concrete structure.

The area of the concrete structure to be painted = 3 – 0.25 kk=4,Area of the concrete structure to be painted = 3 – 0.25(4)

= 2 m2

The shape of the concrete will be a rectangle with length 4m and breadth 0.5m, which may look like this:

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ConclusionA gate is a point of entry to a space enclosed by walls, or a

moderately sized opening in a fence. Gates may prevent or control entry or exit, or they may be merely decorative. Other terms for gate include yett and port.

Larger gates can be used for a whole building, such as a castle or fortified town, or the actual doors that block entry through the gatehouse. Other than that, selection of gate also depends on beautifying or some religious believe in fengshui which brings luck in life.

As I doing this project, I notice that quadratic function and integration can be so close in our life. There are many shape of gate outside there. Different shapes of the gate have different cost. From quadratic function and integration, we can know area of the gate. From the area we can get volume of the gate. As the result, we can know cost of the gate by times volume with RM840 (price of 1m3)

After we know concept of quadratic function and integration, we can apply it in our life. In order to meet the budget or saving money, we can capable to decide on which shape or design more favorable and reasonable.

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Reflection

After spending countless hours,days and night to finish this project and also sacrificing my time for chatting and movies  in this mid year holiday,there are several things that I can say...

Additional Mathematics...From the day I born...From the day I was able to holding pencil...From the day I start learning...And...From the day I heard your name...

I always thought that you will be my greatest obstacle and rival in excelling in my life...But after countless of hours...Countless of days...Countless of nights...

After sacrificing my precious time just for you...Sacrificing my play Time..Sacrificing my Chatting...Sacrificing my Facebook...Sacrificing my internet...Sacrifing my Anime... Sacrificing my Movies...I realized something really important in you...

I really love you...You are my real friend...You my partner...You are my soulmate...

I LOVE U ADDITIONAL MATHEMATIC 28

Reference http://en.wikipedia.org/wiki/ Integral

http://en.wikipedia.org/wiki/Arch

http://www.facebook.com/#!/pages/We-Can-Do-Maths/122938361050773?ref=ts

Additional Mathematics Text Book Form 5

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