add math project 2
TRANSCRIPT

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SEKOLAH MENENGAH KEBANGSAAN TUNKU ABDUL
RAHMAN NIBONG TEBAL
Jalan Bukit Panchor,
14300 Nibong Tebal.
Seberang Perai Selatan,
Pulau Pinang.
PROJECT WORK FOR ADDITIONAL MATHEMATICS 2010
PROJECT WORK 1
SCHOOL : SEKOLAH MENENGAH KEBANGSAAN TUNKU ABDUL
RAHMAN
CLASS : 5 USM
TEACHER : PN. ZUWARIYAH BINTI ABDUL WAHAB
NAME : LIM YEE THING
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I/C : 930619 07 5358
TABLE OF CONTENT
No Title Page Number1 Introduction 32 Acknowledgement 43 History 54 Objective 65 Task specification 7126 Further exploration 13177 Conclusion 188 Reflection 19
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INTRODUCTION
Math is the real world, its everywhere.
I get this sentence from the famous American Series NUMB3RS.
Mathematics is the study of quantity, structure, space and change.
Mathematicians seek out patterns, formulate new conjectures, and
establish truth by rigorous deduction from appropriately chosen
axioms and definitions. We use mathematics to solve simple
mathematic equation and compare the price of product in our daily
life. In fact, mathematics knowledge has been applied in various
fields such as astronomy, business, computing, architecture and the
list goes on. In the NUMB3RS, famous American Series, it shows that
mathematics can be used to predict criminal activities.
Furthermore, mathematics is also being used in the construction
of buildings. With mathematics, we can determine the height of
building, the stability of the building against wind resistance or
earthquake, and the shape of the building which is the most ideal
and unique. Besides, we can calculate the cost needed to construct
a building. Mathematics also helps us to save up cost of
construction by determining the shape of the building which built.
An architect must have this knowledge in order to predict and build
the building exactly the cost that is required to spend.
We can eventually do many things that are beyond our
imagination if we further explore the great mystery of mathematics.
Therefore I am very sure that I will certainly attain and learn
something which benefits me in daily life.
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ACKNOWLEDGEMENT
First of all, I would like to thank sincerely to my parents for
providing me everything that is required to complete this project.For instance, they give me money to buy anything related to this
project work, their advice, and facilities such as computers, books,
and reference material. My parents greatly supported and
encouraged me in order to build my confidence to do this task so
that I will not procrastinate in doing it.
Furthermore, I would like to say thank you to my teacher, Pn.
Zuwariyah, for guiding me throughout this project. She taught us
patiently to do the project, without any complaint. Even when I met
some hardship and difficulties, she taught me patiently until I got
what she mean and what must I do. She tried and did not give up
teaching me until I finished the project successfully.
In addition, my friends also helped me a lot while doing this task.
Although this project is individual project, we discussed and
cooperated to get more information about this project in order to
understand more and make the project more perfect. Our
cooperation does really help each of us a lot in completing this task.
Lastly, those who helped me in this project indirectly, I would
really want to appreciate all of you. Thank you.
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HISTORY
The product rule and chain rule, the notion of higher
derivatives, Taylor series, and analytical functions were introducedby Isaac Newton in an idiosyncratic notation which he used to solve
problems of mathematical physics. In his publications, Newton
rephrased his ideas to suit the mathematical idiom of the time,
replacing calculations with infinitesimals by equivalent geometrical
arguments which were considered beyond reproach. He used the
methods of calculus to solve the problem of planetary motion, the
shape of the surface of a rotating fluid, the oblate ness of the earth,
the motion of a weight sliding on a cycloid, and many other
problems discussed in his Principia Mathematics. In other work, he
developed series expansions for functions, including fractional and
irrational powers, and it was clear that he understood the principles
of the Taylor series.
These ideas were systematized into a true calculus of
infinitesimals by Gottfried Wilhelm Leibniz, who was originally
accused of plagiarism by Newton. He is now regarded as an
independent inventor of and contributor to calculus. His contribution
was to provide a clear set of rules for manipulating infinitesimal
quantities, allowing the computation of second and higher
derivatives, and providing the product rule and chain rule, in their
differential and integral forms. Unlike Newton, Leibniz provided were
the laws of differentiation and integration, second and higher
derivatives, and the notion of an approximating polynomial series.
By Newtons time, the fundamental theorem of calculus was known.
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OBJECTIVE
The aims of carrying out this project work are:
1. To provide learning environment that stimulates and enhances
effective learning.
2. To apply and adapt a variety of problemsolving strategies to
solve problems.
3. To promote effective mathematical communication.
4. To use the language of mathematics to express mathematical
ideas precisely.
5. To develop positive attitude towards mathematics.
6. To develop mathematical knowledge through problem solving
in a way that increases students interest and confidence.
7. To improve thinking skills.
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TASK SPECIFICATION
The diagram below shows the gate of an art gallery. A concrete
structure is built at the upper part of the gate and the words ARTGALLERY is written on it. The top of the concrete structure is flat
whereas the bottom is parabolic in shape. The concrete structure is
supported by two vertical pillars at both ends.
The distance between the two pillars is 4 metres and the height of
the pillar is 5 metres. The height of the concrete structure is 1
metre. The shortest distance from point A of the concrete structure
to point B, that is the highest point on the parabolic shape, is 0.5
metres.
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PROBLEM SOLVING
(a)The parabolic shape of the concrete structure can be representedby various functions depending on the point of reference. Based
on different points of reference, obtain at least three different
functions which can be used to represent the curve of this
concrete structure.
For function 1:
Maximum point (0,4.5) and pass through point (2,4)
b=0, c=4.5
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Substitute (2,4) into (1)
For function 2:
Since (0, ) is the maximum point of the equation,
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Substitute point (2, 0) into equation 2,
Function 2:
For function 3:
Maximum point (2,4.5) and pass through point (0,4)
b=2, c=4.5
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Substitute (0,4) into (3)
(b)The front surface of this concrete structure will be painted before
the words ART GALEERY is written on it. Find the area to be
painted.
Using Function 1,
Area of shaded region
= Area of painted region
= Area of rectangle Area of curve from origin
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= (4 x 1) 
= 4 
= 4 
= 4 
= 4 
= 4  1
= 2
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FURTHER EXPLORATION
(a)You are given four different shapes of concrete structure as
shown in the diagrams below. All the structures have the same
thickness of 40 cm and are symmetrical.
STRUCTURE 1 STRUCTURE 2
STRUCTURE 3 STRUCTURE 4
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(i) Given that the cost to construct 1 cubic metre of concrete is
RM840.00, determine which structure will cost the minimum to
construct.
Structure 1Ignoring the pillars,
Area = 2
Volume = 2 x 0.4
=1
Cost needed =1 x RM840
=RM896.00
Structure 2
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Ignoring the pillars,
Area = 4(0.5)+ 2( )(2)(0.5)
= 3
Volume = 3 x 0.4
=1.2
Cost needed =1.2 x RM840
=RM1008.00
Structure 3Ignoring the pillars,
Area = 4(0.5)+ 1.5(0.5)
= 2.75
Volume = 2.75 x 0.4
=1.1
Cost needed =1.1 x RM840
=RM924.00
Structure 4Ignoring the pillars,
Area = 4(0.5)+ 1(0.5)
= 2.5
Volume = 2.5 x 0.4
=1
Cost needed =1 x RM840
=RM840.00
Therefore, structure 4 will cost minimum to construct.
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(ii) As the president of the Arts Club, you are given the opportunity
to decide on the shape of the gate to be constructed. Which shape
would you choose? Explain and elaborate on your reasons for
choosing the shape.
Answer:
As the president of the Arts Club, I will choose the shape as the
structure 4. Owing to the fact that the cost needed to build the
whole structure is cheaper than the other three concrete structure.
Assuming that the height of pillars is constant for all the four
structures, the cost which is used to construct the shape like the
structure 4 is only RM 840.00 while the other 3 structures cost
higher than structure 4 that is RM840.00.
(b)The following questions refer to the concrete structure in the
diagram below. If the value of k increase with a common
difference of 0.25 m;
(i) Complete Table 1 by finding the values ofkand the
corresponding areas of the concrete structure to be painted.
k(m) Area to be painted ( )
(correct to 4 decimal places)
0.00 3.00000.25 2.93750.50 2.87500.75 2.8125
1.00 2.75001.25 2.6875
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1.50 2.62501.75 2.56252.00 2.5000
(ii) Observe the values of the area to be painted from Table 1. Oyou see any pattern? Discuss.
There is a pattern in the area to be painted.
The area to be painted decreases as the k increases 0.25m and form
a series of numbers:
3, 2.9375, 2.875, 2.8125, 2.75, 2.6875, 2.625, 2.5625, 2.5
We can see that the difference between each term and the next
term is the same.
2.9375 3 = 0.0625
2.875 2.9375 = 0.0625
2.8125 2.875 = 0.0625
2.75 2.8125= 0.0625
2.6875 2.75= 0.0625
2.625 2.6875 = 0.0625
2.5625 2.625 = 0.0625
2.5 2.5625 = 0.0625 Therefore, we can deduce that this series of numbers is an
Arithmetic Progression (AP), with a common difference, d = 0.0625
In conclusion, when k increases 0.25m, the area to be painted
decreases by 0.0625
(iii) Express the area of the concrete structure to be painted in
terms ofk. Find the area when kapproaches the value of 4
and predict the shape of the concrete structure.
Let area to be painted = A
A = 2 +
A = 2 + 1 
A = 3 
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When kapproaches 4,
A = 3 
A = 2
The shape of the concrete structure will be a rectangle as when k=
4, the value of k is the same as the length of upper part of the
concrete structure.
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Conclusion
A gate is a point of entry to a space enclosed by walls, or a
moderately sized opening in a fence. Gates may prevent or controlentry or exit, or they may be merely decorative. Other terms for
gate include yett and port.
Larger gates can be used for a whole building, such as a castle or
fortified town, or the actual doors that block entry through the
gatehouse. Other than that, selection of gate also depends on
beautifying or some religious believe in fengshui which brings luck
in life.
As I doing this project, I notice that quadratic function and
integration can be so close in our daily life. There are many shape of
gate outside there. Different shapes of the gate have different cost.
From quadratic function and integration, we can know area of the
gate. From the area we can get volume of the gate. As the result,
we can know cost of the gate by times volume with RM840 (price for
1m)
After we know concept of quadratic function and integration,
we can apply it in our life. In order to meet the budget or saving
money, we can capable to decide on
which shape or design more favorable and reasonable.
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REFLECTION
While conducting this project, a lot of information that I found. I
have learnt how to build a concrete gate structure with good quality
and proper price.
Apart from that, this project encourages the student to work
together and share their knowledge. It is also encourage student to
gather information from the internet, improve thinking skills and
promote effective mathematical communication.
Last but not least, I proposed this project should be continue
because it brings a lot of moral values to the student and also test
the students understanding in Additional Mathematics.
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