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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 RAHMAN CLASS TEACHER NAME :

:

SEKOLAH MENENGAH KEBANGSAAN TUNKU ABDUL

:

5 USM : PN. ZUAWARIYAH

LIM YEE THING

1

I/C

:

930619 07 5358 Table of Content

No 1 2 3 4 5 6

Title Introduction Task Specification Problem Solving Further Explosion Conclusion Reflection

Page Number

2

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.

4

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 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 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.

Problem Solving

(a) The parabolic shape of the concrete structure can be represented10

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, Substitute point (4, 0) into equation 1,

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Function 1:

For function 2:

Since (0, ) is the maximum point of the equation, Substitute point (2, 0) into equation 2,

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Function 2: For function 3:

Since (-2, ) is the maximum point of the equation, Substitute point (-4, 0) into equation 3,

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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 painted.

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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=4=4-1

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=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 =RM896.00 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 x RM840

Structure 3

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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 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 the19

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.

(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) 0.00 0.25 0.50 0.75 1.00 1.25

Area to be painted ( 3.0000 2.9375 2.8750 2.8125 2.7500 2.6875

)

(correct to 4 decimal places)

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(ii)

1.50 2.6250 1.75 2.5625 2.00 2.5000 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