92572607 additional mathematics project 2012
DESCRIPTION
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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