Addmath Project Work 1 2010

8/8/2019 Addmath Project Work 1 2010

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Assessor : En. Azizan bin Dahaman


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Appreciation 4

Objectives 6Introduction 8

Part 1 10

Part 2 (Further Exploration) 16

Reflection 26


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APPRECIATION

Alhamdulillah, I have finally finished my Additional Mathematics Project Work.

An appreciation goes to my family for their ongoing support and encouragement.

I also would like to thank En. Azizan b. Dahaman and Pn. Zaliana bt. Mat Zain

for their advises and guidance when I was carrying out this project work.

Finally, thanks to my friends for lending me their hands and everyone that had

helped me to complete this task.


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OBJECTIVES

Together with the completion of the Additional Mathematics Project Work, students are

to gain valuable experiences and able to :

y apply and adapt a variety of problem solving strategies to solve routine and non-

routine problems,

y experience classroom environments and hence improve their thinking skills,

y acquire effective mathematical communication through oral and writing, and to

use the language of mathematics to express mathematical ideas correctly and

precisely,

y realize that mathematics is an important and powerful tool in solving real-life

problems and hence develop positive attitude towards mathematics.


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INTRODUCTION

An arch is a structure with a curved top and two straight sides. It is made of

voussoirs, which are wedge-shaped blocks. The topmost voussoir is called a keystone. To

hold an arch together, all of its parts is needed. To construct an arch, a frame known a

centre or centring is built. The voussoirs are placed on it until the arch is completely

formed.

Arch building began from ancient times as early as 2 000 B.C. in Mesopotamian

brick architecture. The Romans then evaluated this technique and uses it for a variety of

structures. The beauty of this design ensures its legacy to the modern times.

The arch comes in many shapes, such as semicircular arch in Rome, segmental

arch by the Chinese and pointed arch founded in Europe. In Middle East, horseshoe-

shaped arches and stacked arches are developed by the Muslim civilization.

al-Aqsa MosqueRoman aqueduct

Wembley Stadium

Angkor Wat


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PART 1

a) Various functions representing the parabolic shape of the concrete structure

Function 1

f(x) = a(x-2) + 4.5 ---(1)

Substitute x = 0, f(x) = 4 into (1)

4 = a(-2) + 4.5

4a = -0.5

a = -0.125

Substitute a = -0.125 into (1)

f(x) = -0.125(x-2) + 4.5

= -0.125 x - 4x + 4 + 4.5

= -0.125x + 0.5x - 0.5 + 4.5

f(x) = -0.125x + 0.5x + 4


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Function 2

f(x) = a(x-2) - 0.5 ---(1)

Substitute x = 0, f(x) = -1 into (1)

-1 = a(-2) - 0.5

4a = -0.5

a = -0.125


f(x) = -0.125(x-2) - 0.5

= -0.125 x - 4x + 4 - 0.5

= -0.125x + 0.5x - 0.5 0.5

f(x) = -0.125x + 0.5x -1


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Function 3

f(x) = ax + 0.5 ---(1)

Substitute x = 2, f(x) = 0 into (1)

0 = a(2) + 0.5

4a = -0.5

a = -0.125


f(x) = -0.125x + 0.5


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b) Area to be painted

From the second function obtained at (a), f(x) = -0.125x + 0.5x -1,

the area to be painted = | area of the coloured region | .

Area of coloured region

=

=

=

-8

3 + 4 4

=-8

3

Area to be painted

=-8

3

=8

3

= 22

3m

= 2.6667 m

( )dx


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FURTHER EXPLORATION

a) i) Since all the structures have the same thickness of 40 cm and every cubic metre

costs RM840.00, the thickness and the cost can be omitted from the calculation.

The lower the area, the lower the cost.

It can be determined that

y The structures in ascending order of costs are Structure 4, Structure 1,

Structure 3 and Structure 2.

y The structure that will cost minimum to construct is Structure 4.

Area of Structure 1

= 2.6667 m

Area of Structure 2

= 2 1

2 2 (1 + 0.5)

= 3 m

Area of Structure 3

= 2 1

2 1.5 (1+ 0.5) + (1 0.5)

= 2.75 m

Area of Structure 4

= 2 1

2 1 (1 + 0.5) + (2 0.5)

= 2.5 m


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To check,

Cost = area thickness price per cubic metre

Cost of Structure 1

=8

3 0.4 840

= RM 896.00

Cost of Structure 2

= 3 0.4 840

= RM 1 008.00

Cost of Structure 3

= 2.75 0.4 840

= RM 924.00

Cost of Structure 4

= 2.5 0.4 840

= RM 840.00

Structure 4 that will cost minimum to construct, which is RM 840.00


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ii) As the president of the Arts Club, I will choose the shape of Structure 1 to be the

shape of the gate to be constructed. Its cost is the second cheapest after

Structure 4 and its difference with the cost of Structure 4 is only RM 56.00.

Although it is not as cheap as Structure 4, it is more attractive. The arch shape

like Structure 1 is also stronger and this is proven by many works of architecture

such as the Roman Coliseum, a structure that is built hundreds of years ago and is

still present today.

The Roman Coliseum


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b) i) Let the vertices of the structure be E, F, G, Q, H, I, P and J.

Assuming that

y the structure is symmetrical,

y JEF,EFG,FGH andEJI are right angles, and

y EJP and FGQ are straight lines,

then the area to be painted consists ofEFGHIJ and identical right angle

triangles HGQ andJIP as shown in figure below.

JIHG = EFJI + IH + HG = 4

JI + HG = 4 - IH

IH = k

JI + HG = 4 - k

HG = JI

2JI = 4 - k

JI =4 - k

2


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The area to be painted is12 - k

4with k increases with a common difference of

0.25 m as shown in Table 1.

k (m) Area to be painted (m)

0.00 3.0000

0.25 2.9375

0.50 2.8750

0.75 2.8125

1.00 2.7500

1.25 2.68751.50 2.6250

1.75 2.5625

2.00 2.5000

Table 1

AEFGQHIPJ = AEFGHIJ + AHGQ + AJIP

= AEFGHIJ + 2AJIP

= (4 0.5) + 2 1

2

4 - k

2 0.5

= 2 +4 - k

4

=12 - k

4


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ii) The graph shows the area to be painted with changing value of k.

From the graph, it can be observed that

y Area to be painted decreases as k increases.

y The area forms an arithmetic progression with the first term 3 and a

common difference -0.0625 m.

y The area decreases with a value of 0.0625 m with every 0.25 m increase

of k.

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

0 0.5 1 1.5 2 2.5

Areatobepainted(m)

k (m)


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c) The area to be painted can be expressed as12 - k

4.

The area when k approaches the value of 4

The area when k approaches the value of 4 equals the area of EFGHIJ. The

concrete structure will be a rectangle with 4 m in length and 0.5 m of breadth as

shown below.

0. 0.5 m

4 m

m


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REFLECTION

Addmath Project Work 1 2010

Documents

Transcript of Addmath Project Work 1 2010