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SEKOLAH MENENGAH KEBANGSAAN TUNKU

ABDUL RAHMAN NIBONG TEBAL

Jalan Bukit Panchor, 14300 Nibong Tebal. S. P. S.

Pulau Pinang.

PROJECT WORK FOR ADDITIONAL MATHEMATICS 2010

PROJECT WORK 1

SCHOOL : SEKOLAH MENENGAH KEBANGSAAN TUNKU ABDUL

RAHMAN

CLASS : 5 USM

TEACHER : PN. ZUAWARIYAH

NAME : LIM YEE THING

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I/C : 930619 – 07 – 5358

Table of Content

No Title Page Number

1 Introduction

2 Task Specification

3 Problem Solving

4 Further Explosion

5 Conclusion

6 Reflection

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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

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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, Teacher

Tan Ru See, 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

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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 introduced

by 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

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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 Newton’s 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 ‘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.

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Problem Solving

(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.

For function 1:

Since (2, ) is the maximum point of the equation,

——①

Substitute point (4, 0) into equation 1,

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Function 1:

For function 2:

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

——②

Substitute point (2, 0) into equation 2,

Function 2:

For function 3:

Since (-2, ) is the maximum point of the equation,

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——③

Substitute point (-4, 0) into equation 3,

Function 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

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painted.

Using Function 1,

Area of shaded region

= Area of painted region

= Area of rectangle – Area of curve from origin

= (4 x 1) -

= 4 -

= 4 -

= 4 -

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= 4 -

= 4 - 1

= 2

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FURTHER EXPLOTRATION

(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 1

Ignoring the pillars,

Area = 2

Volume = 2 x 0.4

=1

Cost needed =1 x RM840

=RM896.00

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Structure 2

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 3

Ignoring 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 4

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Ignoring the pillars,

Area = 4(0.5)+ 1(0.5)

= 2.5

Volume = 2.5 x 0.4

=1

Cost needed =1 x RM840

=RM840.00

Therefore, structure 4 will cost minimum to construct.

(ii) As the president of the Arts 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.

Answer:

As the president of the Arts Club, I will choose the shape as the

structure 4. Owing to the fact that the cost needed to build the

whole structure is cheaper than the other three concrete structure.

Assuming that the height of pillars is constant for all the four

structures, the cost which is used to construct the shape like the

structure 4 is only RM 840.00 while the other 3 structures cost

higher than structure 4 that is RM840.00.

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(b)The following questions refer to the concrete structure in the

diagram below. If the value of k increase with a common

difference of 0.25 m;

(i) Complete Table 1 by finding the values of k and the

corresponding areas of the concrete structure to be painted.

k(m)Area to be painted ( )

(correct to 4 decimal places)

0.00 3.0000

0.25 2.9375

0.50 2.8750

0.75 2.8125

1.00 2.7500

1.25 2.6875

1.50 2.6250

1.75 2.5625

2.00 2.5000

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(ii) Observe the values of the area to be painted from Table 1. O

you see any pattern? Discuss.

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

2.6875 – 2.75 = -0.0625

2.625 – 2.6875 = -0.0625

2.5625 – 2.625 = -0.0625

2.5 – 2.5625 = -0.0625

Therefore, we can deduce that this series of numbers is an

Arithmetic Progression (AP), with a common difference, d = -0.0625

In conclusion, when k increases 0.25m, the area to be painted

decreases by -0.0625

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Conclusion

After doing a lot of researches, answering questions, drawing

graphs and some problem solving, I saw that the usage of calculus is

very important in our daily life. It is not just widely used in science,

economics but also in engineering. As a conclusion, calculus is a

daily life necessity. Without is, marvellous buildings can’t be built,

human beings will not lead to a luxurious life and many more. So, we

should be thankful of the people who had contributed the idea of

calculus.

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Reflection

While I was conducting this project, I had learned some

important moral values that I practice. This project had taught me to

responsible on the works that are given to me to be completed. This

project also had made me felt more confident to do works and not to

give up easily when we could not find the solution for the question. I

also learned to be more discipline on time, which I was given about a

month to complete this project and pass up to my teacher just in

time. I also enjoy doing this project during my school holiday as I

spend my time with friends to complete this project and it had

tightened our friendship too.

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