the complete answer of first question of add math project 2012

7/28/2019 The Complete Answer of First Question of Add Math Project 2012

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Problem Solving

Pak Samy has a piece of unused land besides his house. This piece of land is surrounded by

the river and the mountain. After his retirement, he decided to clear up that piece of land to

plant vegetables. The outlook of the land is as shown in the Diagram 1.

Pak Samy thinks that it will be good if he can fence up that land. He measured the diagonal

distance from the river to the food of the mountain (A toB) is 500m and the distance along the

mountain side till it almost meets the streams of the river (B to C) is 800m. Pak Samy also built a

block made from sand bags along the river for flood prevention during heavy rain. The angle

subtended between the diagonal distance of AB and the sand block is 30 degree as shown in

Diagram 2.


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a) Pak Samy planned to dig a well with cross section of the shape of a sector with centre point B,

to make watering job easier for him. He needs to build the top part of the well with radius 1m

and height 1m. You are required to help him to calculate the angle ofABC in order that he could

build the well. Provide at least two methods in your solutions.

Understanding of problem: Find the angle ofABC by using two different solutions.

Strategy planning: 1) Using the sine rule

2) Using the draw and measure method

Carrying out strategy:

Strategy 1: Sine rule

BC ag = AB dff

sin BAC sin ACB

800 ag = 500 dff

gsin 30 sin ACB

ACB = 18.21

Since it is a triangle, total internal angle = 180

ABC = 180- 18.21 -30

= 131.79


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Strategy 2: Draw and measure method

Note: The diagram is drawn using geogebra with correct scale.

Using protractor, it is measured that ABC = 131.79

Comment: Based on the sine rule and the drawn triangle, angle of 131.79 is required for

ABC so that Pak Samy is able to dig a well with cross section shape of a sector with

centre point B.


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(b) Pak Samy wants to build the thickest wall for the well. You are supposed to help him to

calculate the number of bricks to buy in order to build the well. As shown in diagram 3. For the

curve surface, calculation should base on the internal surface area. Given 400cm2 = 4 bricks

inclusive of cement in between bricks. If each brick is 40 cents each, help Pak Samy to calculate

how much he needs to spend on bricks.

Understanding of problem: Find the cost of bricks to be used for building the wall of the well.

Strategy planning: Simple calculation of area of walls of bricks and multiplication


1) Calculating the length of arc of curved surface,

Since ABC = 131.79= 2.3 rad.

Using the formula:

S=r

Where S=length of curved surface, r=radius of circle, = ABC in rad

S= 100cm x 2.3 rad.

= 230cm

2) Calculating the area and the cost

Since height is 1m= 100cm,


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Total surface area= (230x100)+(100x100)+(100x100)

= 23000+10000+10000= 43000cm2

Total bricks needed:

43000 x 4 = 430 brick

h400

Amount needed to spend on bricks: 430 x 0.40= RM172

Comment: To build the walls, 430 bricks are required. The cost for building the wall is RM172.


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c) Pak Samy is poor in calculation, he wanted to fence up that piece of land in a triangular shape.

You need to help him to calculate the total length of the fencing materials needed. Use at least 2

methods of solution.

Understanding Problem: Calculate the total length of the fencing materials needed to fence up

the land in triangular shape.

Strategy planning: 1) Using cosine rule

2) Draw and measure method


Strategy 1: Cosine rule

Since ABC = 131.79,

AC2

=AB2

+BC2

- 2(AB)(BC) cos ABC

AC2 = 5002 +8002 -2(500)(800) cos 131.79

AC2 = 1423121.88

AC= 1192.95m

Total length need= AB+AC+BC= 500+1192.95+800= 2492.95m


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Strategy 2: Draw and measure method using graph

Using geogebra,

Triangle with correct scales on graph is drawn as shown in below,

Using point A and C,

Distance = (x1-x2) + (y1-y2)

Area=(596.47)2 + (1033.12)2

= 1192.94 units

= 1192.94m

Total length of fencing= 800+500+1192.94= 2492.94m

Comment: Total length of 2492.95m of materials is required to fence up the piece of land in a

triangular shape.


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d) After clearing up and fencing the land, he needs to buy seeds of vegetables and plants is so

that he could buy sufficient of various seeds. As an additional mathematics student, you need to

help him to find the area of the land by using at least 2 different methods.

Understanding of problem: Find the area of the land needed so Pak Samy can buy sufficient

amount of various seeds for his vegetables and plants.

Strategy planning: 1) Using formula of area of triangles

2) Using area of polygon of coordinates on graph


Strategy 1: Area of triangles

Area of triangle

=(AB)(BC)sin ABC

=(500)(800)sin 131.79

=149118m2


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Strategy 2:Area of polygon of coordinates on graph

1stcoordinate= A (0,-500)2nd coordinate= B (0,0)

3rd coordinate= C (596.47,533.12)

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

= |(0 + 0-298235 0 0 0)|

= |(0 - 298235)|

= 149117.5

= 149118 unit2 = 149118 m2

Comment: The area is plotted to identify the coordinates of the points of the triangle. These

solutions show that the area of the land is 149118 m2 .


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e) After sometimes, the fence is rundown, Pak Samy wishes to build the new fence with

minimum cost. He wishes to minimize the fencing materials but the area of the piece of land

must remain the same. The point B and C are fixed because they are by the mountain side. Only

point A is movable. Make a conjecture on the position of point A. (Tabulate a few sets of values of

AB and AC, and find the minimum length of the fence.)

Understanding of problem: Find the minimum length of the fence to determine the postion of

point A.

Strategy planning: By using a table to find the minimum length of the fence.


AB ABC AC Total length

400 68.75 753.68 1953.68

450 55.94 662.76 1912.76

500 131.79 1192.95 2492.95

550 42.67 543.56 1893.56

600 38.41 497.75 1897.75

650 35 458.89 1908.89

700 32.18 426.68 1926.68

750 29.81 402 1,952

800 27.77 383.96 1983.96

850 26.01 374.49 2024.5

900 24.47 373.29 2073.29

From the table,

The minimum length of AC and AB are 550m and 543.56m respectively. So the minimum length

of the fence is 1893.56m. So the conjectures are made.

Comment: From the table, minimum length of 1893.56m of fence is required for Pak Samy to

minimize the cost of fencing materials and the area of the piece of land still remain

fixed.


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f) A year later, seeing the price of oil palm is so attractive. Pak Samy is planning to divide the

land into two parts, he wants to plant oil palm which is the golden plant at the farther part of the

land. He had another piece of fencing material that is same length with the length of AB. He

wants to build the fence from point B outside the well in a straight line until it reaches the lineAC. You are required to help him to construct the location of the fence in graphical form. He

decided to differentiate the color of the two fences for the two types of plants beside the river;

you are asked to help Pak Samy to find the length of the fences of vegetable and oil palm.

Understanding of problem: Divide the land of Pak Samy into two parts to plant oil palm at the

further part of the land.

Planning strategy: Using graphical form to draw the divided lands by different colours.


The graphical form of diagram is drawn with correct scale as shown in below,


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From the graph, the light brown area of land is used to plant oil palm while the pink area is used

to plant Pak Samys vegetables

Using point A and E,

AE = (x1-x2) + (y1-y2)

Area=(433.01)2 + (750)2

=866 units

= 866m

Using point E and C,

EC = (x1-x2) + (y1-y2)

Area=(163.46)2 + (283.12)2

= 326.92 units

= 326.92m

Length of fence for vegetables= AB+AE+BE= 500+500+866= 1866m

Length of fence for oil palm= BC+BE+EC= 800+500+326.92=1626.93m

Comment: Using the geogebra, forming graphical form, the total length of fences of vegetables

and oil palm are 1866m and 1626.93m respectively.


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g) Pak Samy hires some workers to clear the piece of land to plant oil palm. The workers ask for

RM60 per 400m2 as wages. At the end of the clearing process, Pak Samy pays a total sum of

RM6126.40 to the workers as wages. Find the area of the land that Pak Samy uses to plant oil

palm.

Understanding of problem: Calculate the area of the land that Pak Samy uses to plant oil palm.

Planning Strategy: Using simple multiplication and dividation to solve.

Carrying out strategy;

Since they pay total sum of RM6126.40,

RM6126.40 X 400 =40842.67m2

RM60

Comment: The area that Pak Samy uses to plant oil palm is 40842.67m2 .


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Further exploration:

After 5 years the oil palm trees are mature and ready for harvest, Pak Samy finds that the

return is so attractive. Now he wishes to convert the whole land to oil palm. He wishes to build

more solid fence for the whole land. He bought 2000m of fencing materials and the side by the

mountain side is fixed (BC), find the dimension of triangle ABC such that the area enclosed ismaximum so that he could plant the most oil palm trees. Hence, find the maximum area of the

plantation.

Understanding of problem: Find dimension of triangle ABC such that area enclosed is maximum,

thus find the maximum area.

Strategy Planning: Using Herons formula.


Based on the Herons formula,

Heron's formula for the area of a triangle with sides of length a, b, c is

where

Let the c be fixed BC that is 800m, AB be a, AC be b,

Since the total length is fixed that is 2000m and BC=800m,

The maximum total length of AB and AC would be 1200m

Tabulation of length of AB and AC,

AB AC BC Total area

600 600 800 178885.44

700 500 800 173205.08

400 800 800 154919.33

900 300 800 118321.6


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Objectives:

a) To apply mathematics to everyday situations and appreciate the importance and thebeauty of mathematics in everyday lives.

b) To improve problem-solving skills, thinking skills, reasoning and mathematicalcommunication.

c) To develop positive attitude and personalities and intrinsic mathematical values suchas accuracy, confidence, and systematic reasoning.

d) To stimulate learning environment that enhances effective learning, inquiry-based andteamwork.

e) To develop mathematical knowledge in a way that increases students interest andconfidence.


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Appreciation

First of all, I would like to say thank you to my friends, teachers and parents for giving me

their full support in making this project successful.

Not forgotten to my family for providing everything, such as money, to buy anything that

are related to this project work and their advise, which is the most needed for this project.

Internet, books, computers and all that act as my source to complete this project. They also

supported me and encouraged me to complete this task so that I will not procrastinate in

doing it.

Then I would like to thank my teacher, Mr. Johnattan James for guiding me and my friends

throughout this project. We had some difficulties in doing this task, but he taught us

patiently and gave me guidance throughout the journey until we knew what to do. He tried

his best to help us until we understand what we supposed to do with the project work.

Besides that, my friends who were doing this project with me and shared our ideas. Theywere helpful that when we combined and discussed together, we had this task completed in

just a glimpse of an eye.

Last but not least, any party which involved either directly or indirect in completing this

project work.


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Reflection

Throughout the project while I was conducting it, I learned many stuff. This includes on

usage of knowledge and ways to conduct the project. While I was conducting the project, I

collected information from the internet and reference books regarding the triangles.

However, the most important this I have learnt from this project is to plan before we do

something. In this project, I am required to determine the area of the land for Pak Samy to

plant vegetable. The length of the fence required is also determined for the piece of land

beside his house. Angle is also be calculated in order to build a well for easier watering job.

While answering, I made few tables on the angles and the length of sides to minimise the

fencing materials of the length. But the area of the piece of land must be fixed so cost can be

minimize. While finding the maximum area for Pak Samy to plant oil palm trees, this

tabulation method is also used to make a conjecture.

Besides, I learned how to cooperate with friends. My friends and I discussed about the

project and we shared ideas amongst ourselves. This discussion has made me more

confident when doing something. I also learned to be disciplined in terms of punctuality,

doing project and not least to avoid plagiarism.

In a nutshell, I have learned something about the consturction of land and how serious can

it be if we plan it wrongly. Moreover, work that we done must be original and not totally

derived from other people. These two images have expressed my opinions from this project.


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Introduction

In polygon, triangle is a polygon with three sides with a total sum of each angle of 180

degree.Triangles have been the most popular and common geometric form for planningplantation since the shape is easy to stack and organize; as a standard, it is easy to design a

plan or map to fit. But rectangles while more difficult to use conceptually, provide a great

deal of strength. As computer technology helps us to design creative land, triangular shapes

are becoming increasingly prevalent to shape a piece of land for planting various plants or

vegetables that can give a maximum profit.

Law of Sines

According to Ubiratn D'Ambrosio and Selin Helaine, the spherical law of sines was

discovered in the 10th century. It is variously attributed to al-Khujandi, Abul Wafa

Bozjani, Nasir al-Din al-Tusiand Abu Nasr Mansur.

Al-Jayyani's The book of unknown arcs of a sphere in the 11th century introduced the

general law of sines.[ The plane law of sines was later described in the 13th century byNasr

al-Dn al-Ts. In his On the Sector Figure, he stated the law of sines for plane and spherical

triangles, and provided proofs for this law.

The formula is proved to be true!
http://en.wikipedia.org/wiki/Ubirat%C3%A0n_D%27Ambrosiohttp://en.wikipedia.org/wiki/Abu-Mahmud_al-Khujandihttp://en.wikipedia.org/wiki/Abul_Wafa_Bozjanihttp://en.wikipedia.org/wiki/Abul_Wafa_Bozjanihttp://en.wikipedia.org/wiki/Nasir_al-Din_al-Tusihttp://en.wikipedia.org/wiki/Abu_Nasr_Mansurhttp://en.wikipedia.org/wiki/Al-Jayyanihttp://en.wikipedia.org/wiki/Nas%C4%ABr_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Nas%C4%ABr_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Nas%C4%ABr_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Nas%C4%ABr_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Nas%C4%ABr_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Nas%C4%ABr_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Nas%C4%ABr_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Nas%C4%ABr_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Nas%C4%ABr_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/Al-Jayyanihttp://en.wikipedia.org/wiki/Abu_Nasr_Mansurhttp://en.wikipedia.org/wiki/Nasir_al-Din_al-Tusihttp://en.wikipedia.org/wiki/Abul_Wafa_Bozjanihttp://en.wikipedia.org/wiki/Abul_Wafa_Bozjanihttp://en.wikipedia.org/wiki/Abu-Mahmud_al-Khujandihttp://en.wikipedia.org/wiki/Ubirat%C3%A0n_D%27Ambrosio


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According to Glen Van Brummelen, "The Law of Sines is

really Regiomontanus's foundation for his solutions of right-

angled triangles in Book IV, and these solutions are in turn the

bases for his solutions of general triangles." Regiomontanus

was a 15th-century German mathematician.

Regiomontanus

Herons formula

Heron

Heron's (or Hero's) formula, named after Heron of Alexandria, states that the area T of

a triangle whose sides have lengths a, b, and c is

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

where s is the semi-perimeter of the triangle:

Heron of Alexandria was an ancient Greek mathematician and engineer who was active in

his native city ofAlexandria, Roman Egypt. He is considered the greatest experimenter of

antiquityand his work is representative of the Hellenistic scientific tradition.

Heron's formula is distinguished from other formulas for the area of a triangle, such as half

the base times the height or half the modulus of a cross product of two sides, by requiring

no arbitrary choice of side as base or vertex as origin.
http://en.wikipedia.org/wiki/Glen_Van_Brummelenhttp://en.wikipedia.org/wiki/Regiomontanushttp://en.wikipedia.org/wiki/Heron_of_Alexandriahttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Trianglehttp://en.wikipedia.org/wiki/Semiperimeterhttp://en.wikipedia.org/wiki/Greek_mathematicshttp://en.wikipedia.org/wiki/Engineerhttp://en.wikipedia.org/wiki/Alexandriahttp://en.wikipedia.org/wiki/Roman_Egypthttp://en.wikipedia.org/wiki/Hellenistic_civilizationhttp://en.wikipedia.org/wiki/Hellenistic_civilizationhttp://en.wikipedia.org/wiki/Roman_Egypthttp://en.wikipedia.org/wiki/Alexandriahttp://en.wikipedia.org/wiki/Engineerhttp://en.wikipedia.org/wiki/Greek_mathematicshttp://en.wikipedia.org/wiki/Semiperimeterhttp://en.wikipedia.org/wiki/Trianglehttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Heron_of_Alexandriahttp://en.wikipedia.org/wiki/Regiomontanushttp://en.wikipedia.org/wiki/Glen_Van_Brummelen


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SMK Chung Hua Miri

ADDITIONAL MATHEMATICS

PROJECT WORK 1/2012

Lets Explore theTriangles

Name : Ngu Leon Weiy

I/C No : 951102-13-6413

Class : 5S1

Teachers Name : Mr. Johnattan James


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Content:

1) Appreciation

2) Objectives

3) Introduction

4) Problem Solving

5) Further Exploration

6) Conclusion

7) Reflection


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From Leon HAHAHAHHAHAH

I love JJ!!! Not sure which JJ right????? HAHAHAHA

Remember follow the arrangement of content, live with triangles is

cover de

the complete answer of first question of add math project 2012

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