NAME : AMIRUL NAQIB BIN RAZAK CLASS I/C NUMBER

: :

5 ST 1

TEACHER SCHOOL

::

PUAN NURUL IDZWATY BT MOHD NAZIR SMK BANDAR BARU SALAK TINGGI

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 2

Objectives

The aims of carrying out this project work are:

I. To apply and adapt a variety of problem-solving strategies to solve problems

II. To improve thinking skills

III. To promote effective mathematical communication

IV. To develop mathematical knowledge through problem solving in a way that increases students interest and confidence

V. To use the language of mathematics to express mathematical ideas precisely

VI. To provide learning environment that stimulates and enhances effective learning

VII. To develop positive attitude towards mathematics

ACKNOWLEDGEMENT

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 3

First and foremost, I would like to thank God that finally, I have succeeded in

finishing this project work. I would like to thank my beloved Additional

Mathematics teacher, Pn. Nurul Idzwaty Bt. Mohd Nazir for all the guidance

she had provided me during the process in finishing this project work. I also

appreciate her patience in guiding me completing this project work. I would

like to give a thousand thanks to my father and mother, Razak bin Mohd

Mazlan and Zalina binti Abdul Rahman, for giving me their full support in this

project work, financially and mentally. They gave me moral support when I

needed it. Who am I without their love and support? I would also like to give

my thanks to my fellow friends who had helped me in finding the information

that I’m clueless of, and the time we spent together in study groups on

finishing this project work. Last but not least, I would like to express my

highest gratitude to all those who gave me the possibility to complete this

coursework. I really appreciate all the help I got.

Again, thank you very much.

CONTENTS

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 4

NO CONTENT PAGE

1 OBJECTIVES 2

2 ACKNOWLEDGEMET 3

3 INTRODUCTION 5

4 PART 1

Part 1 (a)

Part 1 (b)

Part 1 (c)

6 7 10 11

5 PART 2

Part 2 (a)

Part 2 (b)

Part 2 (c)

Part 2 (d)(i)

Part 2 (d)(ii)

Part 2 (d)(iii)

13 14 14 14 14 14 14

6 PART 3

Part 3 (a)

Part 3 (b)(i)

Part 3 (b)(ii)

15 16 18 21

7 REFLECTION 22

8 REFERENCES 23

INTRODUCTION

A polygon is a flat shape consisting of straight lines that are joined to form a closed chain or circuit.

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 5

A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. An n-gon is a polygon with n sides. The interior of the polygon is sometimes called its body. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions.

The word "polygon" derives from the Greek πολύς (polús) "much", "many" and γωνία (gōnía) "corner" or "angle". (The word γόνυ gónu, with a short o, is unrelated and means "knee".) Today a polygon is more usually understood in terms of sides.

The basic geometrical notion has been adapted in various ways to suit particular purposes. Mathematicians are often concerned only with the closed polygonal chain and with simple polygons which do not self-intersect, and may define a polygon accordingly. Geometrically two edges meeting at a corner are required to form an angle that is not straight (180°); otherwise, the line segments will be considered parts of a single edge – however mathematically, such corners may sometimes be allowed. In fields relating to computation, the term polygon has taken on a slightly altered meaning derived from the way the shape is stored and manipulated in computer graphics (image generation)

Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, and the pentagram, a non-convex regular polygon (star polygon), appears on the vase of Aristophonus, Caere, dated to the 7th century B.C. Non-convex polygons in general were not systematically studied until the 14th century by Thomas Bredwardine. In 1952, Shephard generalized the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons.

SELANGOR EDUCATION DEPARTMENT

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012

PART 1 Polygons are evident in all architecture. They provide variation and charm in buildings. When applied to manufactured articles such as printed fabrics,

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 6

wallpapers, and tile flooring, polygons enhance the beauty of the structure itself. (a) Collect six such pictures. You may use a camera to take the pictures

or get them from magazines, newspapers, internet or any other resources.

(b) Give the definition of polygon and write a brief history of it. (c) There are various methods of finding the area of a triangle. State four different methods. a)

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 7

The Kaaba is a cuboid-shaped building in Mecca, Saudi Arabia

The Egyptian pyramids are ancient pyramid-shaped masonry structures located in Egypt.

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 8

Contemporary Home Design in Polygon Shape with Marvelous Panorama at the Pittman Dowell Residence

Rectangular shaped bricks

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 9

Pentagon-shaped tiles

Trapezium-shaped house

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 10

b) Definition and History of Polygon :

In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain or circuit.

A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. An n-gon is a polygon with n sides. The interior of the polygon is sometimes called its body. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions.

The word "polygon" derives from the Greek πολύς (polús) "much", "many" and γωνία (gōnía) "corner" or "angle". (The word γόνυ gónu, with a short o, is unrelated and means "knee".) Today a polygon is more usually understood in terms of sides

History.

Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, and the pentagram, a non-convex regular polygon (star polygon), appears on the vase of Aristophonus, Caere, dated to the 7th century B.C Non-convex polygons in general were not systematically studied until the 14th century by Thomas Bredwardine.

In 1952, Shephard generalized the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons.

C) Area of Triangle :

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 11

Method 1

Method 2

Method 3

Method 4

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 12

Area = 1

2 |( + + ) − ( + + )|

PART 2

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A farmer wishes to build a herb garden on a piece of land. Diagram 1 shows the shape of that garden, where one of its sides is 100 m in length. The garden has to be fenced with a 300 m fence. The cost of fencing the garden is RM 20 per metre. (The diagram below is not drawn to scale)

(a) Calculate the cost needed to fence the herb garden. (b) Complete table 1 by using various values of p, the corresponding

values of q and θ.

p (m) q (m) θ (degree) Area (m2)

Table 1

(c) Based on your findings in (b), state the dimension of the herb

garden so that the enclosed area is maximum. (d)(i) Only certain values of p and of q are applicable in this case.

State the range of values of p and of q. (ii) By comparing the lengths of p, q and the given side, determine

the relation between them. (iii) Make generalisation about the lengths of sides of a triangle.

State the name of the relevant theorem.

(a) Cost = RM 20 × 300 = RM 6000.

p ( m) q ( m) θo Area (m2 )

p m q m mm

100 m c Diagram 1

θº

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 14

(b)

Using cosine rule, ab

cba

2cos

222 Area = sin

2

1ab

(c) The herb garden is an equilateral triangle of sides 100 m with a maximum area of 4330.13 m 2 . (d)(i) 50 < p < 150, 50 < q < 150 (ii) p + q ˃ 100 (iii) The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Triangle Inequality Theorem.

In this case, p + q > 100. For the relevant theorem the length of sides can be related to the 𝜃 according to cosinus rule. In order to find , we can refer to :

cos 𝜃 = 1002+𝑝2+𝑞2

−2𝑝𝑞

𝜃 = cos -1(1002+𝑝2+𝑞2

−2𝑝𝑞)

50 150 0 0

60 140 38.2145 2598.15

65 135 44.8137 3092.33

70 130 49.5826 3464.10

80 120 55.7711 3968.63

85 115 57.6881 4130.68

90 110 58.9924 4242.64

95 105 59.7510 4308.42

99 101 59.9901 4329.26

100 100 60 4330.13

p m q m mm

100 m c Diagram 1

θº

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 15

PART 3 If the length of the fence remains the same 300m, as stated in part2 :

(a) Explore and sugest at least 5 various other shapes of the garden that can be constructed so that the enclosed area is maximum.

(b) Draw a conclusion from your exploration in (a) if : (i) The demand of herbs in the market has been increasing nowadays. Suggest three types of local herbs with their scientific names that the farmer can plant in the herb garden to meet the demand. Collect pictures and information of these herbs. (ii) These herbs will be processed for marketing by a company. The design of the packaging plays an important role in attracting customers. The company wishes to design an innovative and creative logo for the packaging. You are given the task of designing a logo to promote the product. Draw the logo on a piece of A4 paper. You must include at

least one polygon shape in the logo.

(a)

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 16

(a) Quadrilateral

x y Area = x y

10 140 1400

20 130 2600

30 120 3600

40 110 4400

50 100 5000

60 90 5400

70 80 5600

75 75 5625

The maximum area is 5625 m2.

(b) Regular Pentagon 5a = 300

a = 60

mt

t

2915.4154tan30

3054tan

273.61935)602915.41(2

1mArea

(c) A Semicircle

(d) A Circle

r m •

r m •

x m

y m 2x + 2y = 300 m2

x + y = 150 m2

Area = x y

a

a a

a

a 72o

54o 54o

t

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 17

(e) A Regular Hexagon

Conclusion: Circle is the best shape to use for the garden as it gives a maximum enclosed area among the other shapes.

2

2

97.7161)2(150

2

1

150

3002

mArea

r

r

(b) FURTHER EXPLORATION (i) 3 Suggested types of herbs:

(i) Cymbopogon

Cymbopogon (lemongrass) is a genus of about 55 species of grasses, (of which the type species is Cymbopogon citratus)

50 m

60º 6a = 300 a = 50

Area =

a

a

a

a

a

a

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 18

native to warm temperate and tropical regions of the Old World and Oceania. It is a tall perennial grass. Common names include lemon grass, lemongrass, barbed wire grass, silky heads, citronella grass,cha de Dartigalongue, fever grass, tanglad, hierba Luisa or gavati chaha amongst many others.

Uses:

Lemongrass is native to India and tropical Asia. It is widely used as a herb in Asian cuisine. It has a subtle citrus flavor and can be dried and powdered, or used fresh.

Lemongrass is commonly used in teas, soups, and curries. It is also suitable for poultry, fish, beef, and seafood. It is often used as a tea in African countries such as Togo and the Democratic Republic of the Congo and Latin American countries such as Mexico.Lemongrass oil is used as a pesticide and a preservative. Research shows that lemongrass oil has anti-fungal properties.

(ii) Orthosiphon stamineus (misai kucing)

Orthosiphon stamineus is a traditional herb that is widely grown in tropical areas. The two general species, Orthosiphon stamineus "purple" and Orthosiphon stamineus "white" are traditionally used to treat diabetes, kidney and urinary disorders, high blood pressure and bone or muscular pain.

Also known as Java tea, it was possibly introduced to the west in the early 20th century. Misai Kucing is popularly consumed as a herbal tea.

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 19

The brewing of Java tea is similar to that for other teas. It is soaked in hot boiling water for about three minutes, and honey or milk is then added. It can be easily prepared as garden tea from the dried leaves. There are quite a number of commercial products derived from Misai Kucing.Sinensetin is a polyphenol found in O. stamineus.

.

(iii) Ficus deltoidea (Mas Cotek)

Mas Cotek (Ficus deltoidea) (in Thai Language) is a tree species native to Malaysia.

Malaysia's tropical rainforest is unique, with a large biodiversity of valuable plants and animals. The discovery of herbal plants in these jungles, and in particular Mas Cotek (Ficus deltoidea), is slowly receiving international recognition for its medicinal values and health benefits. Based on traditional knowledge, the leaves, fruits, stems and roots of Mas Cotek display healing, palliative and preventative properties

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 20

Mas Cotek, also known as "mistletoe fig", has been scientifically researched by various institutions, including University of Malaya, Universiti Putra Malaysia, Universiti Sains Malaysia, the Forest Research Institute of Malaysia, the Malaysian Agriculture Research And Development Institute (MARDI). Research results show that Mas Cotek possesses five classes of chemicals, namely flavonoids, tannins, triterpenoids, proanthocyanins and phenols

Traditionally used as a postpartum treatment to help in contracting the muscles of the uterus and in the healing of the uterus and vaginal canal, it is also used as a libido booster by both men and women.The leaves of male and female plants are mixed in specific proportions to be taken as an aphrodisiac.[ Among the traditional practices, Mas Cotek has been used for regulating blood pressure, increasing and recovering sexual desire, womb contraction after delivery, reducing cholesterol, reducing blood sugar level, treatment of migraines, toxin removal, delay menopause, nausea, joints pains, piles pain and improving blood circulation.

Mas Cotek products are formulated and sold in the form of extracts, herbal drinks, coffee drinks, capsules, and massage oil.

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 21

b) Godiva Lemongrass package

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 22

REFLECTION

While I conducting this project, a lot of information that I found.I have

learnt the uses of polygons. I also learned some moral values that I practice.

This project had taught me to be responsible on the works that are given to

me to be completed. This project also made me felt more confidence 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

three weeks to complete these project and pass up to my teacher just in time.

I also enjoyed doing this project during my school holiday as I spent my time

with friends to complete this project and it had tighten our friendship

REFERENCES

ADDITIONAL MATHEMATICS PROJECT WORK 1/2012 23

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

http://www.scribd.com/

https://www.facebook.com/

Additional Mathematics Text Book