Addmaths Project 2012 - Polygons

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    AdditionalMathematic

    sKINGSTON HERBAL LAND

    Name : Meishaleny David

    Class : 5 Perdana

    Teacher : Puan Maimunah Binti Taib

    IC Number : 950606-10-6408

    Angka Giliran : BH045A013

    School : SMK Seksyen 19, Shah Alam

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    OBJECTIVE

    The aims of carrying out this project are to enable students to:

    Apply mathematics in everyday situations and appreciate the importance

    and the beauty of mathematics in everyday life

    Improve problem solving skills,thinking skills,reasoning and

    mathematical communication

    Develop positive attitudes and personalities intrinsic mathematical values

    such as accuracy,confidence and systematic reasoning

    Stimulate learning environment that enhances effective learning inquiry

    base and teamwork

    Develop mathematical knowledge in away which increases students

    interest and confidence

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    ACTNOWLEDGEMENT

    First of all, I would like to thank my teacher Puan Maimunah @ Asmah

    Binti Taib for giving such a beneficial project and for guiding me and my

    friends throughout this project.

    We did had some difficulties in doing this paperwork ,but teacher had

    taught us patiently the right way to do this project and made us realise our

    mistakes. Not forgotten my both parents, I am really thankful to them for their

    support and encouragement throughout this project and for their advices.

    Last but not least, I would like to thank my friends also for their supports

    and helps. They had helped me a lot and shared their ideas. Thanks to my

    friends once again to spare their time with me to do this project. My friends

    deserve a big part of my project.

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    PART 1

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    (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.

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    (b) Give the definition of polygon and write a brief history of it.

    Definition of polygon

    A closed shape consisting of line segments that has at least three sides. Triangles,

    quadrilaterals, rectangles, and squares are all types of polygons.

    History of polygon

    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) There are various methods of finding the area of a triangle. State four different

    methods.

    Method 1:

    h

    b

    If you know base (b) and height (h) of the triangle, the following formula can be applied.

    Area = x b x h

    Method 2:

    a c

    b

    If you know three sides (a, b and c) of the triangle, Herons Method can be applied.

    s = (a+b+c) / 2

    Area = s (s-a) (s-b) (s-c)

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    Method 3:

    a

    b

    If you know two sides (a and b) and the included angle (), the following formula can be

    applied.

    Area = x a x b x sin

    Method 4:

    If you know coordinate of the three vertices, the following formula can be applied.

    = (x1y2 + x2y3 + x3y1 x2y1 x3y2 x1y3)

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    Method 5:

    If you know equation (y=mx+c) of the three lines, integration can be applied.

    Area = (mx1 + c) dx + (mx2 + c) dx - (mx3 + c) dx

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    PART 2

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    (a) Calculate the cost needed to fence the herb garden.

    Cost needed to fence the herb garden = RM20.00 per metre x 300m

    = RM6000.00

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    (b) Complete table 1 by using various values ofp, the corresponding values ofq and .

    p

    (m)

    q

    (m)cos degree area (m2)

    p

    (m)

    q

    (m)cos degree area (m2)

    50 150 1.0000 0.00 0.00 101 99 0.5002 59.99 4329.26

    51 149 0.9739 13.11 861.68 102 98 0.5006 59.96 4326.66

    52 148 0.9491 18.37 1212.44 103 97 0.5014 59.91 4322.33

    53 147 0.9253 22.29 1477.33 104 96 0.5024 59.84 4316.25

    54 146 0.9026 25.50 1697.06 105 95 0.5038 59.75 4308.42

    55 145 0.8809 28.25 1887.46 106 94 0.5054 59.64 4298.84

    56 144 0.8601 30.67 2056.70 107 93 0.5074 59.51 4287.48

    57 143 0.8403 32.83 2209.64 108 92 0.5097 59.36 4274.34

    58 142 0.8213 34.79 2349.47 109 91 0.5122 59.19 4259.40

    59 141 0.8031 36.57 2478.41 110 90 0.5152 58.99 4242.64

    60 140 0.7857 38.21 2598.08 111 89 0.5184 58.78 4224.04

    61 139 0.7691 39.73 2709.70 112 88 0.5219 58.54 4203.57

    62 138 0.7532 41.14 2814.25 113 87 0.5258 58.28 4181.21

    63 137 0.7379 42.45 2912.47 114 86 0.5300 58.00 4156.92

    64 136 0.7233 43.67 3005.00 115 85 0.5345 57.69 4130.68

    65 135 0.7094 44.81 3092.33 116 84 0.5394 57.36 4102.44

    66 134 0.6961 45.89 3174.90 117 83 0.5446 57.00 4072.16

    67 133 0.6833 46.90 3253.08 118 82 0.5502 56.62 4039.80

    68 132 0.6711 47.85 3327.16 119 81 0.5562 56.21 4005.31

    69 131 0.6595 48.74 3397.43 120 80 0.5625 55.77 3968.63

    70 130 0.6484 49.58 3464.10 121 79 0.5692 55.31 3929.69

    71 129 0.6377 50.38 3527.39 122 78 0.5763 54.81 3888.44

    72 128 0.6276 51.13 3587.48 123 77 0.5838 54.28 3844.80

    73 127 0.6179 51.83 3644.52 124 76 0.5917 53.72 3798.68

    74 126 0.6088 52.50 3698.65 125 75 0.6000 53.13 3750.0075 125 0.6000 53.13 3750.00 126 74 0.6088 52.50 3698.65

    76 124 0.5917 53.72 3798.68 127 73 0.6179 51.83 3644.52

    77 123 0.5838 54.28 3844.80 128 72 0.6276 51.13 3587.48

    78 122 0.5763 54.81 3888.44 129 71 0.6377 50.38 3527.39

    79 121 0.5692 55.31 3929.69 130 70 0.6484 49.58 3464.10

    80 120 0.5625 55.77 3968.63 131 69 0.6595 48.74 3397.43

    81 119 0.5562 56.21 4005.31 132 68 0.6711 47.85 3327.16

    82 118 0.5502 56.62 4039.80 133 67 0.6833 46.90 3253.08

    83 117 0.5446 57.00 4072.16 134 66 0.6961 45.89 3174.90

    84 116 0.5394 57.36 4102.44 135 65 0.7094 44.81 3092.33

    85 115 0.5345 57.69 4130.68 136 64 0.7233 43.67 3005.00

    86 114 0.5300 58.00 4156.92 137 63 0.7379 42.45 2912.4787 113 0.5258 58.28 4181.21 138 62 0.7532 41.14 2814.25

    88 112 0.5219 58.54 4203.57 139 61 0.7691 39.73 2709.70

    89 111 0.5184 58.78 4224.04 140 60 0.7857 38.21 2598.08

    90 110 0.5152 58.99 4242.64 141 59 0.8031 36.57 2478.41

    91 109 0.5122 59.19 4259.40 142 58 0.8213 34.79 2349.47

    92 108 0.5097 59.36 4274.34 143 57 0.8403 32.83 2209.64

    93 107 0.5074 59.51 4287.48 144 56 0.8601 30.67 2056.70

    94 106 0.5054 59.64 4298.84 145 55 0.8809 28.25 1887.46

    95 105 0.5038 59.75 4308.42 146 54 0.9026 25.50 1697.06

    96 104 0.5024 59.84 4316.25 147 53 0.9253 22.29 1477.33

    97 103 0.5014 59.91 4322.33 148 52 0.9491 18.37 1212.44

    98 102 0.5006 59.96 4326.66 149 51 0.9739 13.11 861.68

    99 101 0.5002 59.99 4329.26 150 50 1.0000 0.00 0.00

    100 100 0.5000 60.00 4330.13

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

    enclosed area is maximum.

    In order for the enclosed area to be maximum, the shape of the triangle must be

    equilateral. Therefore, the length of both p and q must be 100m.

    (d)

    (i) Only certain values ofp and ofq are applicable in this case. State the range of values

    ofp and ofq

    50 < p

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    PART 3

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    (a) Explore and suggest at least 5 various other shapes of the garden that can be

    constructed so that the enclosed area is maximum.

    1. Quadrilateral

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    2. Regular Pentagon

    3. Regular Hexagon

    Area = (33/2) x rx r

    = 2861.7856m

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    4. Semicircle

    5. Circle

    (b) Draw a conclusion from your exploration in (a) if :

    i) The demand of herbs in the market has been increasing nowadays. Suggest 3 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.

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    Aloe Vera - The local name is known as 'Lidah Buaya'. Due to its exceptional healing

    properties, Aloe Vera is also known as "the little of the desert", "the plant of immortality",

    "the first aid plant" and "the miracle plant". It is also one of the most important crude drug

    history and is still extensively used in modern medicine. It contains the same painkilling

    and anti-inflammatory compound as in Aspirin. Aloe vera is also a nutrient-rich-plant,

    containing more than 200 active-components, vitamins, minerals, essential amino acids,enzymes and other plant chemicals which is said to gently strengthen, sustain and

    encourage cellular activity in body.

    Tongkat Ali - It has become popular for its testosterone-enhancing properties. Because of

    that, it is included in certain herbal supplements for bodybuilders. Historically, it has been

    used by the folk medicine in its countries of origin as a libido enhancer and to treat various

    sexual dysfunctions. Numerous scientific studies performed by Malaysian Universities,

    including University Science Malaysia (USM) have confirmed its effects on enhancing

    sexual characteristics in animal models. Currently, it is being researched for its possibilities

    as an anti-cancer supplement.

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    Agrimonia - commonly known as Agrimony. It is not commonly used today, but has its

    place in traditional herbal medicine. This herb is safe for use for minor ailments in most

    healthy people. Like most herb simples, the uses to which it is put are remarkably varied.

    The English use it to make a delicious "spring" or "diet" drink for purifying the blood. It is

    considered especially useful as a tonic for aiding recovery from winter colds, fevers, and

    diarrhea. Agrimony contains tannin and a volatile essential oil.

    ii) These herbs will be processed for marketing by a company. The design of a 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 size A4. You must include at

    least one polygon shape in the logo.

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