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SEKOLAH MENENGAH KEBANGSAAN TUNKU
ABDUL RAHMAN NIBONG TEBAL
Jalan Bukit Panchor, 14300 Nibong Tebal. S. P. S.
PROJECT WORK FOR ADDITIONAL MATHEMATICS 2010
PROJECT WORK 1
SCHOOL : SEKOLAH MENENGAH KEBANGSAAN TUNKU ABDUL
CLASS : 5 USM
TEACHER : PN. ZUAWARIYAH
NAME : LIM YEE THING
I/C : 930619 – 07 – 5358
Table of Content
No Title Page Number
2 Task Specification
3 Problem Solving
4 Further Explosion
“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.
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, Teacher
Tan Ru See, 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.
The product rule and chain rule, the notion of higher
derivatives, Taylor series, and analytical functions were introduced
by 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 Newton’s time, the fundamental theorem of calculus was known.
The aims of carrying out this project work are:
1. To provide learning environment that stimulates and enhances
2. To apply and adapt a variety of problem-solving strategies to
3. To promote effective mathematical communication.
4. To use the language of mathematics to express mathematical
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.
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 ‘ART
GALLERY’ 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
(a)The parabolic shape of the concrete structure can be
represented by 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:
Since (2, ) is the maximum point of the equation,
Substitute point (4, 0) into equation 1,
For function 2:
Since (0, ) is the maximum point of the equation,
Substitute point (2, 0) into equation 2,
For function 3:
Since (-2, ) is the maximum point of the equation,
Substitute point (-4, 0) into equation 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
Using Function 1,
Area of shaded region
= Area of painted region
= Area of rectangle – Area of curve from origin
= (4 x 1) -
= 4 -
= 4 -
= 4 -
= 4 -
= 4 - 1
(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
(i) Given that the cost to construct 1 cubic metre of concrete is RM840.00, determine which structure will cost
the minimum to construct.
Ignoring the pillars,
Area = 2
Volume = 2 x 0.4
Cost needed =1 x RM840
Ignoring the pillars,
Area = 4(0.5)+ 2( )(2)(0.5)
Volume = 3 x 0.4
Cost needed =1.2 x RM840
Ignoring the pillars,
Area = 4(0.5)+ 1.5(0.5)
Volume = 2.75 x 0.4
Cost needed =1.1 x RM840
Ignoring the pillars,
Area = 4(0.5)+ 1(0.5)
Volume = 2.5 x 0.4
Cost needed =1 x RM840
Therefore, structure 4 will cost minimum to construct.
(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.
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 of k and the
corresponding areas of the concrete structure to be painted.
k(m)Area to be painted ( )
(correct to 4 decimal places)
(ii) Observe the values of the area to be painted from Table 1. O
you 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
After doing a lot of researches, answering questions, drawing
graphs and some problem solving, I saw that the usage of calculus is
very important in our daily life. It is not just widely used in science,
economics but also in engineering. As a conclusion, calculus is a
daily life necessity. Without is, marvellous buildings can’t be built,
human beings will not lead to a luxurious life and many more. So, we
should be thankful of the people who had contributed the idea of
While I was conducting this project, I had learned some
important moral values that I practice. This project had taught me to
responsible on the works that are given to me to be completed. This
project also had made me felt more confident to do works and not to
give up easily when we could not find the solution for the question. I
also learned to be more discipline on time, which I was given about a
month to complete this project and pass up to my teacher just in
time. I also enjoy doing this project during my school holiday as I
spend my time with friends to complete this project and it had
tightened our friendship too.