SMK SERI TANJONG

45000 KUALA SELANGOR

ADDITIONAL MATHEMATICS

PROJECT WORK 2015

NAME : DEVENDRAN S/O KANABATHY

I/C NO. : 980531-10-6063

CLASS : 5 SYUKUR

TEACHER NAME : PN.MAZIATUL LISA BINTI MAHMOOD

CONTENTS

ACKNOWLEGDEMENT

OBJECTIVE

INTRODUCTION

PART 1

PART 2

PART 3

FURTHER EXPLORATION

REFLECTION

ACKNOWLEDGEMENT

My name is Devendran s/o Kanabathy . I am thankful that this Additional Mathematics Project can be done just in time. For this, I would like toseize the opportunity to express my sincere gratitude for those who had been helping me during my work.

First and foremost, I would like tosay a big thank you to my Additional Mathematics teacher, Pn. Maziatul Lisa Binti Mahmood for giving me information about my project work. On the other hand, I would also like tothank my dear principle, Yang Mulia Puan Hajah Asmah Binti Kasdi for giving me the permission to carry out this project.

Also, I would like to thank my parents. They had brought me the things that I needed during the project work was going on. Not only that, they also provided me withthe nice suggestion on my project work so that I had not meet the dead and throughout this project.

Lastly, I would like to say thank you to my friends and the modern access in our daily life. All of my relevant information come from my friends and the internet. I managed to use all these access in our daily life, such as: computer to finish my Additional Mathematics project.

OBJECTIVE

Additional Mathematics is one of the compulsory subjectsfor SPM science stream candidates. All of the students would have to carry out a project work based on a topic given and must be submitted in three weeks time.

The objective of carrying out thisproject is:

To apply and adapt a variety of problem-solving strategies that we had learnt to solvethe problems.

Our thinking skills can be improved.

Promotes effective mathematical communication. Our confidence and interest towards Mathematics will beincrease though solving various types of problems.

To use the language ofMathematics to express Mathematical ideas precisely.

Stimulates and enhances effective earning.

To develop our positive attitude towards Mathematics. This makes the lesson to be more fun, useful andmeaningful.

ADDITIONAL MATHEMATICS PROJECT WORK

INTRODUCTION

In mathematics, the maximum and minimum point of function, known collectively as extrema are the largest and smallest value that the function takes at a point either within a given neighbourhood (local or relative extremum) or on the function domain in its entirely ( global or absolute extremum ). Pierre de Fermat was one of the first mathematicians to propose a general technique (called adequality) for finding maxima and minima. To locate extreme values is the basic objective of optimization.

PART 1

(a)

(i) Mathematical optimization deals with the problem of finding numerically minimums (or maximums or zeros) of a function. In this context, the function is called cost function, or objective function, or energy.

(ii) In mathematical analysis, the maxima and minima (the plural of maximum and minimum) of a function, known as collectively as extrema (the plural of extremum), are the largest and smallest value of the function within the entire domain of a function (the global or absolute extrema).

We say that f(x) has an absolute (or global) maximum at x=c if f(x) f (c) for every x in the domain we are working on.

We say that f(x) has an absolute (or global) minimum at x=c if if f(x) f (c) for every x in the domain we are working on.

(iii) In mathematical analysis, the maxima and minima (the plural of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, within a given range (the local of relative extrema)

We say that f(x) has a relative (or local) maximum at x=c if f(x) < f (c) for every x in some open interval around.

We say that f(x) has relative (or local) minimum at x=c of f(x) > f (c) for every x in some open interval around.

(b)

Small change

Max & min value

Tangent and normal equation

Differentation

Turning points

Completing the squares

Axis of symmetry

Rate of change

Draw graphs

Type of turning point

- second derivative (dy/dx)

create new quadratic equation after reflection

Part 2

a)

b) REZAS BOX

Part 3

i)

Based on the equation, a table has been constructed.

t / hours

P / number of people

0

0

1.5

527

3

1800

4.5

3073

6

3600

7.5

3073

9

1800

10.5

527

12

0

ii) When does the mall reach the mall reach its peak hours and state the number of people

9.30 a.m + 6 hours

= 3.30 p.m

By assuming that the mall opens during business hours 9.30 a.m, the peak hours

is at 3.30 p.m with the number of people approximately 3600 people

iii) Estimate the number of people in the mall 7.30 p.m

The number of the people into the mall at 7.30 p.m is about 900 people

iv) Determine the time when the number of people in the mall 7.30 reaches 2570

It is 1.18 p.m when the number of people reaches 12570

Further Exploration

Linear programming (LP; also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization).

More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of infinitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine function defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest (or largest) value if such a point exists.

The first programming formulation of a problem that is equilavent to the general linear programming problem was given by Leonid Kantorovich in 1939, who also proposed a method for solving it. He developed it during World War II as a way to plan expenditures and returns so as to reduce costs to the army and increase losses incurred by the enemy. Linear programming is the process of taking various linear inequalities relating to some situation, and finding the best value obtainable under those conditions. The general process for solving linear-programming exercise is to graph the inequalities (called the constraints) to form a walled-off area on the x,y-plane (called the feasibility region). Schools and universities throughout the country offer linear programming studies. Because linear programming can be used in various professional and study fields. According to the Stanford University Computer Science Department, the model of linear programming can be applied in many study field, including those listed below.

1)Distribution

2)Transportation

3)Telecommunication

4)Agriculture

5)Production

b)

Cabinet x

Cabinet y

Cost (RM)

100

200

Space ( m )

0.6

0.8

Volume ( m )

0.8

1.2

100 x + 200y < 1400Ratio

x + 2y < 14x : y > 2 : 3

Let x + 2y = 14x > 2/3

When x = 0, y = 7 (0,7)y.

When y = 0 , x = 14 (14,0)3x > 2y

2y < 3x

0.6 x + 0.8 y < 7.2 (x=10) Let 2y = 3x

6 x + 8 y < 72 ( 2) when x = 0, y = 0 3x + 4y < 36so = (0,0)

Let 3x + 4y = 36

When x = 0, y = 9 (0,9) when x = 6, y = 9

When y = 0, x = 12 (12,0) so, (6,9)

The 3 inequalities which satisfy all the constraints.

(i) x + 2y < 14

(ii) 3x + 4y < 36

(iii) 2y < 3x

Method 1

Volume = 0.8 x + 1.2 y ( Maximum point 8.3 )

the equation is 0.8 x + 1.2 y = 1.92

When x = 0,

0.8 (0) + 1.2 y = 1.92

1.2 y = 1.920.8 x 1.2 x 2 = 1.92

y = 1.92/1.2

y = 1.6 so, ( 0,16 )

When y = 0,

0.8 x + 1.2 (0) = 1.92

0.8x = 1.92

x = 2.4 so, ( 2.4, 0 )

From the graph the maximum storage volume = 8 (0.8) + 3 (1.2)

= 10m

Method 2

Point

Cabinet x

Cabinet y

Total Volume = 0.8 x + 12y

(4.5)

4

5

0.8 (4) + 1.2 (5) = 9.2

(5.4)

5

4

0.8 (5) + 1.2 (4) = 8.8

(6.4)

6

4

0.8 (6) + 1.2 (4) = 9.6

(7.3)

7

3

0.8 (7) + 1.2 (3) = 9.2

(8.3)

8

3

0.8 (8) + 1.2 (3) = 10

(9.2)

9

2

0.8 (9) + 1.2 (2) = 9.6

(10.1)

10

1

0.8 (10) + 1.2 (1) = 9.2

From the table above the maximum storage volume is 0.8 (8) + 1.2 (3) = 10 m

(iii)

Cabinet x

Cabinet y

Total cost (RM)

4

6

1600

5

5

1500

6

4

1400

7

3

1300

8

3

1400

9

2

1300

(iv) Aaron should buy the combination of 8 cabinet x and 3 cabinet y. This combination suits his allocation of RM1400 as other combination can be too much. This combination has the largest volume of 10 m compare to the other combination which has less volume generated. It also fulfill the ratio of cabinet x to cabinet y more than 2:3.

REFLECTION

While I conducting this project, a lot of information that I found. I have learnt how maximum and minimum appear in our daily life. 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. Not only that, I had learned some 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 confidence to do works and not to give 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 I spend my time with friends to complete this project and it had strengthen our friendship. Last but not least, I proposed this project should be continued because it brings a lot of moral value to the student and also test the students understanding in Additional Mathematics. Let me end this project with a poem;

In math you can learn everything,

Like maybe youll like comparing,

You have to know subtraction,

a.k.a brother of addition,

You might say I already simplified,

so now your work aint jankedified,

So now dont think negative,

Its better to think positive,

Dont stab yourself with a fork,

But its better to show your work,

My math grades are fat,

But not as fat as my cat,

Lets get typical,

And use a pencil,

Add Math is fun!