ADDITIONAL MATHEMATICS PROJECT WORK 2012NAME CLASS : ZARUL AZHAM BIN MOHD ZAKI : 5 LUKISAN KEJUTERAAN I/C NUMBER : 950928 - 10 - 5299 TEACHER SCHOOL : PUAN LILI FARIZA MAJID : SMK BANDAR BARU SALAK TINGGI Page 1 OBJECTIVES The aims of carrying out this project work are: I. II. III. IV. To apply and 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 increases student interest and confident To use the language of mathematics to express mathematical ideas precisely. To provide learning environment that stimulates and enhance effective learning To develop positive attitude towards mathematics. V. VI. VII. Page 2 ACKNOWLEDGEMENT 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, Puan Lili Fariza Majid 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 my mother, Mohd Zaki bin Mohamed and Junita binti Mohd Said, for giving me their full support in this project work, financially and mentally. To 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 and finishing the project work. Last but not least, I would like to express my highest gratitude who all those who gave me the possibility to complete this courswork. I really appreciate all the help I got. Again thank you very much. Page 3 CONTENT NO CONTENT PAGE 1. OBJECTIVES 2. AKCNOWLEGDEMENT 3. INTRODUCTION 4. PART 1 Part 1 (a) Part 1 (b) Part 1 (c) 5. PART 2 Part 2 (a) Part 2 (b) Part 2 (c) Part 2 (d)(i) Part 2 (d)(ii) Part 2 (d)(iii) 6. PART 3 Part 3 (a) Part 3 (b)(i) Part 3 (b)(ii) 7. REFLECTION 8. REFERENCE Page 4 INTRODUCTION 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. 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. Page 5 DEPARTMENT OF EDUCATION SELANGOR Additional Mathematics Project Work 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, 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. Page 6 a) The Kaaba is a cuboid-shaped building in Mecca, Saudi Arabia The Egyptian pyramids are ancient pyramid-shaped masonry stuctures located in Egypt. Page 7 Contemporary Home Design in Polygons Shape with Marvellous Panorama at the Pittman Dowell Residence Rectangular shaped bricks Pentagon-shaped tiles Trapezium-shaped house Page 8 b) Definition and History of Polygons: 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: 1 base height 2 1 2. Area a b sin C 2 1. Area 3. Area s( s a)(s b)(s c) , where s 4. Area 1 (a b c) , or the semi-perimeter. 2 1 x1 2 y1 x2 y2 x3 y3 x1 y1 Page 9 PART 2 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) p m θº q m mm 100 m c Diagram 1 (a) (b) Calculate the cost needed to fence the herb garden. 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. Page 10 (a) (b) Cost = RM 20 × 300 = RM 6000. First method a m θº b m mm 100 m c Diagram 1 Using cosine rule, cos a2 b2 c2 2ab 1 Area = ab sin 2 p ( m) 50 60 65 70 80 85 90 95 99 100 q ( m) 150 140 135 130 120 115 110 105 101 100 θo 0 38.2145 44.8137 49.5826 55.7711 57.6881 58.9924 59.7510 59.9901 60 Area (m2 ) 0 2598.15 3092.33 3464.10 3968.63 4130.68 4242.64 4308.42 4329.26 4330.13 (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. Page 11 PART 3 If the length of the fence remains the same 300m, as stated in part2: a) Explore and suggest 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 herb in the market has been increasing nowadays. Suggest three types of local herb with their scientific names that the farmer can plant in the herb garden to meet the demand. Collect pictures and information of these herb. 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. Page 12 PART 3 (a) Quadrilateral (suggested answer) 2x + 2y = 300 m2 ym xm x + y = 150 m2 Area = x y x 10 20 30 40 50 60 70 75 y 140 130 120 110 100 90 80 75 Area = x y 1400 2600 3600 4400 5000 5400 5600 5625 The maximum area is 5625 m2. (b) Regular Pentagon 5a = 300 a = 60 t 30 t 30 tan 54 41 .2915 m 1 Area (41.2915 60) 5 6193.73 m 2 2 tan 54 a a a 54o 72o t 54o a a (c) A Semicircle rm • (d) A Circle rm Page 13 • (i) Serai Wangi (Cymbopogon nardus ) Lemongrass is native to India and tropical Asia. It is widely used as a herb in Asian cuisine. It has a subtle citrus flavour 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. Cymbopogon citratus from the Philippines, where it is locally known as tanglad. Page 14 (ii) Misai Kucing (Orthosiphon stamineus) 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. 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. Page 15 (iii) Mas cotek (Ficus deltoidea) Mas Cotek (Ficus deltoidea) ( Malaysia. in Thai Language) is a tree species native to 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. 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. Page 16 PERFECTLY NATURAL HERBS Page 17 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 value that I practice. This project had taught me to be responsible on the works that are given to me to be complete. This project also made me felt more confidence to works and not to give up easily when we could not find the solution for the question. I also learned to be disciplined on time, which I was given three week to complete these projects and pass up to my teacher just in time. Page 18 REFERENCES http://en.wikipedia.org/wiki/Polygon http://www.scribd.com/ https://www.facebook.com/ Additional Mathematic Text Book Page 19