Mine Addmath

7/31/2019 Mine Addmath

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-Content-

No. Contents Page

1 Introduction 3

2 Acknowledgement 4

3 Task Specification 5

4 Problem Solving ( a ) 7

5 ( b ) 9

6 ( c ) 10

7 ( d ) 12

8 ( e ) 13

9 ( f ) 1510 ( g ) 16

8 Further Exploration 17

9 Conclusion 18

10 Reflection 19


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IntroductionWe students taking Additional Mathematics are required to carry out a project work while

we are in Form 5. We are given two tasks and asked to choose and complete only ONE task based on

our area of interest. This project can be done in groups or individually. Upon completion of theAdditional Mathematics Project Work, we are to gain valuable experiences and able to :

Apply and adapt a variety of problem solving strategies to solve routine and non-routine problems

Experience classroom enviroments which are challenging, interesting andmeaningful and hence improve their thinking skills

Experience classroom enviroments where knowledge and skills are applied inmeaningful ways in solving real-life problems

Experience classroom enviroments where expressing ones mathematical thinking,reasoning and communication are highly encouraged and expected

Experience classroom enviroments that stimulates and enhances effective learning

Acpuire effective mathematical communication through oral and writing, and to usethe mathematics to express mathematical ideas correctly and precisely

Enhance acquisition of mathematical knowledge and skills through problem-solvingin ways that increase interest and confidence

Prepare ourselves for the demand of our future undertakings and in workplace Realise that mathematics is an important and powerful tool in solving real-life

problems and hence develop positive attitude towards mathematics

Train ourselves not only to be independent learners but also to collaborate, tocooperate, and to share knowledgein an engaging and healthy environment

Use technology especially the ICT appropriately and effectively Train ourselves to appreciate the intrinsic values of mathematics and to becomemore creative and innovative Realize the importance and the beauty of mathematics


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Aknowledgement

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, Mr Jason Lee for all

the guidance he had provided us during the process in finishing this project work.

I also appreciate his patience in guiding us completing this project work. i would like to give a

thousand thanks to my parent for giving me their full support in this project work, financially and

mentally. They gave me moral support when I needed it. I would also like to give my thanks to my

fellow friends who had helped me in finding the information that I clueless of, and the time I 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.


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Task SpecificationADDITIONAL MATHEMATICS PROJECT WORK 1/2012

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 foot of the mountain (A to B) is 500 m and the distance along the

mountain side till it almost meets the streams of the river (B to C) is 800 m. 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 as shown in diagram 2.

(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 1 m andheight 1 m. You are required to help him to calculate the angle of in order that he could build the

(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 400 = 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. 2 cm

(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.

(d) After clearing up and fencing the land, he needs to buy seeds of vegetables and plants. However,

before he could buy those seeds, he needs to know how big the land 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 two different methods.

(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.)

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

to two parts, he wants to plant oil palm which is the golden plant at the farther part of the land. Hehad another piece of fencing material that is same length with the length of AB. He wants to build


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the fence from point B outside the well in a straight line until it reaches the line AC. 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.

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

60 per 400 as wages. At the end of the clearing process, Pak Samy pays a total sum of RM 6126.40 to

the workers as wages. Find the area of the land that Pak Samy uses to plant oil palm. 2 m

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 plant oil palm. He wishes to build more

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

side is fixed (BC), find the dimension of triangle ABC such that the area enclosed is maximum so that

he could plant the most oil palm trees. Hence, find the maximum area of the plantation.


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METHOD 2

B 800m C

500m

30 D

A

Draw a perpendicular line from B to AC to intersect the line AC at D.

ABD = 60

In triangle ABD

Sin 30 =

BD = 500 sin 30

= 250m

In triangle BDC

Cos DBC =

DBC =

= 71 47

ABC = 60 + 71 47

= 131 47 / 131.79


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b)

The length of internal arc

= (

= 2.3m

= 230cm

Total surface area of the internal arc = 230 x 100

= 23000cm

Area of the 2 squared walls = 100 x 100 x 2

= 20000cm

Total area = 23000+ 20000

= 43000cm

The thickest wall by the dimension of one brick,

Given 400cm = 4 bricks

Total bricks required = (43000/400) x 4

= 430 bricks.

Given cost of one bricks is 40 cents

Total cost required = 430 x 0.4

= RM 172.00


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c)

METHOD 1

B 800m C

500m

30 D

A

Sin 30 =

BD = 250

In triangle ABD, AD = 500 - 250

= 187500

AD =

= 433.01m

In triangle BDC, CD = 800 - 250

= 577500

CD =

= 759.93m

Total length of the fence = 500 + 800 + 433.01 + 759.93

= 2492.94m


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METHOD 2

In triangle ABC

Using Cosine Rule,

AC = 500 + 800 - 2 x 500 x 800 x cos 131 47

= 1 423 052.47

AC =

= 1 192.92m

Total length of fences = 500 + 800 + 1192.92

= 2492.92m


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d)

METHOD 1

Area of the land =

x AC x BD

=

x 1192.92 x 250

= 149115m

METHOD 2

Area of the land =

= 149118m


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e)

Conjecture: The length of the fence is minimum when triangle ABC is an isosceles triangle,

where AB = AC and BC = 800m.

Area of triangle ABC = 149134m

149134 =

x 800 x h

= 372.84m

B x D 800-x C

372.84m

A

Let AD be the height of triangle ABC with BC as the corresponding base.

AD = 372.84m

Let BD = x m

Therefore, DC = 800 x m


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BD DC AB AC Perimeter

100 700 386.01 793.10 1979.11

200 600 423.10 706.41 1929.51

300 500 478.55 623.71 1902.26

400 400 546.82 546.82 1893.64

500 300 623.71 478.55 1902.26600 200 706.41 423.10 1929.51

700 100 793.10 386.01 1979.11

397 403 544.6271 549.0161 1893.6432

398 402 545.3565 548.2825 1893.6390

399 401 546.0867 547.5497 1893.6364

400 400 546.8178 546.8178 1893.6356

401 399 547.5497 546.0867 1893.6364

402 398 548.2825 545.3565 1893.6390

403 397 549.0161 544.6271 1893.6432

Perimeter

minimum

From the tables above, the perimeter is minimum when AB = AC = 546.8178m

399.7 400.3 546.5983 547.0373 1893.635602

399.8 400.2 546.6715 546.9641 1893.635559

399.9 400.1 546.7446 546.8909 1893.635534

400 400 546.8178 546.8178 1893.635525

400.1 399.9 546.8909 546.7446 1893.635534

400.2 399.8 546.9641 546.6715 1893.635559

400.3 399.7 547.0373 546.5983 1893.635602


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f)

B 800m C

500m

30 AA A A'C

A

Length of vegetable fence ( AA )

=

AA = 866.0254m.

Length of Oil Palm fence ( AC )

AC = 326.92m

Or AC AA = 1192.9176 866.0254

= 326.8922m


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g)

Wages RM 60 for 400m

Total wages paid is RM 6126.40

Area of the land = ( 6126.40 / 60 ) x 400

= 40 842.7m


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Further Exploration :-

B 800m C

A

Let BC = 800m and ABC =

The perimeter, AB + BC + AC = 2000m

AB AC Area of 300 900 99.59 118322

400 800 75.52 154919500 700 60 173205

600 600 48.19 178885

650 550 43.05 177482

750 450 33.56 165831

850 350 24.25 139642

Area maximum

From the table of values, it is noted that for the area is maximum when triangle is an isosceles

triangle with BC = 800m and AB = AC = 600m

The maximum area of triangle ABC is 178 885m

597 603 48.51078 178880.407

598 602 48.40357 178883.2021

599 601 48.29654 178884.8792

600 600 48.18969 178885.4382

601 599 48.08301 178884.8792

602 598 47.97651 178883.2021

603 579 47.87019 178880.407

559.7 600.3 48.22172 178885.3879

559.8 600.2 48.21104 178885.4158

599.9 600.1 48.20036 178885.4326

600 600 48.18969 178885.4382

600.1 599.9 48.17901 178885.4326

600.2 599.8 48.16834 178885.4158

600.3 599.7 48.15766 178885.3879


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ConclusionAfter doing research, answering questions, drawing table and some problem solving, I saw

that the usage of triangle is important in our daily life. It is not just widely used in architecture but

also in interpreting the area of a specific location. Especially in measuring a location of area which tobe built something or else.

Without it, all this measurement activities cant be conducted accurately. So, I should

thankful of the people who contribute in the idea of making this way of measurement.


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Reflection

Additional Mathematics..

I struggle so hard to finish you

You taught me the meaning of calculating

You taught me the way to think

Everything is about you

Additional Mathematics..

Your question was so hard to answer

But I survive this storm and rain just to answer you

I swim all this river to get to you

You are everything..

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