PART 1 (a) (i)

An optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a defined domain (or a set of constraints), including a variety of different types of objective functions and different types of domains.

Given: a function f : A   R from some set A to the real numbersSought: an element x0 in A such that 

☻f(x0) ≤ f(x) for all x in A (minimization) ☻such that f(x0) ≥ f(x) for all x in A (maximization)

(a)(ii)

Global (or Absolute) Maximum and Minimum

The maximum or minimum over the entire function is called an "Absolute" or "Global" maximum or minimum.

There is only one global maximum (and one global minimum) but there can be more than one local maximum or minimum.

Assuming this function continues downwards to left or right:

The Global Maximum is about 3.7 The Global Minimum is −Infinity

(a)(iii)

Local Maximum and Minimum

Functions can have "hills and valleys": places where they reach a minimum or maximum value.

It may not be the minimum or maximum for the whole function, but locally it is.

Local Maximum

First we need to choose an interval:

Then we can say that a local maximum is the point where:

The height of the function at "a" is greater than (or equal to) the height anywhere else in that interval.

Or, more briefly:

f(a) ≥ f(x) for all x in the interval

In other words, there is no height greater than f(a).

Note: f(a) should be inside the interval, not at one end or the other.

Local Minimum

Likewise, a local minimum is:

f(a) ≤ f(x) for all x in the interval

PART 2

(a)

y

x

y

Answer:4x+2y= 2002x+y= 100y= 100-2x

x = - b2a = −1002(−2) = 25 m

When x = 25 m, substitute x in y

y = 100-2(25) = 50 m

MAXIMUM AREA =

25m x 50m = 1250 m2

1

2

x x x

(b)

30-2h Area= xy

Substitute in

Area= x(100-2x) = 100x-2 x2 = -2 x2+100x

a = -2 b = 100

30-2h

Volume = L x W x H (length x width x height)

= (30−2h)2 x h

biggest volume = maximum value = (30−2h)2(900-60h-60h+4 h2)h(4 h2-120h+900)h

isipadu = 4 h3-120h2+900h

dvdx = 0

V’ = 12h2-240h+900 = 0(h-5)(h-15) = 0 h = 15h = 5

when h = 15, when h = 5,4(15)3-120(15)2 + 900(15) 4(5)3 -120(5)2 + 900(5)= 0 (minimum) = 2000 cm3 (maximum)

2 1

Largest possible volume : 2000 cm3

PART 3

(a)(i)

X 0 1 2 3 4 5 6 7 8 9 10 11 12

Y 0 241

900

1800

2700

3359

3600

3359

2700

1800

900

241

0

Based on the graph,

(ii) The number of people at 3.30 pm are 3600. (iii) The number of people in the mall at 7.30 pm (iv) The time when the people in the mall reaches 2570 are 1:21 pm and 5:39 pm. are 900.

FOREWARD

I am thankful that this Additional Mathematics Project can be done just in time. For this, I would like to seize the opportunity to express my sincere gratitude for those who had been helping me during my work.

First and foremost, I would like to say a big thank you to my Additional Mathematics teacher, Puan Ruslina Binti Ahmad for giving me information about my project work. We had some difficulties in doing this task, but she taught us patiently until we knew what to do. She tried and tried to teach us until we understand what we supposed to do with the project work. On the other hand, I would also like to thank my principle, Tuan Haji Addenan Bin Osman for give 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 with the 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 beloved friends and the modern access in our daily life. They were helpful that when we combined and discussed together, we had this task done. 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.

FOREWARD

I am thankful that this Additional Mathematics Project can be done just in time. For this, I would like to seize the opportunity to express my sincere gratitude for those who had been helping me during my work.

First and foremost, I would like to say a big thank you to my Additional Mathematics teacher, Puan for giving me information about my project work. We had some difficulties in doing this task, but she taught us patiently until we knew what to do. She tried and tried to teach us until we understand what we supposed to do with the project work. On the other hand, I would also like to thank my dear principle, Puan Siti Dinar for give 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 with the 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. They were helpful that when we combined and discussed together, we had this task done. 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.

CONCLUSION

After doing research, answering question, drawing graph, making conjecture, conclusion and some problem solving, I realize that Additional Mathematics is very important in daily life. This project also helped expose the techniques of application of additional mathematics in real life situations. While conducting this project, a lot of information that I found.

About this project, overall, is quite joyful and interesting because I need to plan it carefully and systematic because it is about my future. 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. In fact, the further exploration is a good session because it can open my mind about linear programming and quadratic function.

In a nutshell, I can apply the concept and skills that I had in solving problems in Additional Mathematics, I think this project is very beneficial for all students. 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.

CONTENTS

BIL. CONTENTS

1 OBJECTIVE

2 FOREWARD

3 INTRODUCTION

4 PART 1

5 PART 2

6 PART 3

7 FURTHER EXPLORATION

8 REFLECTION

9 CONCLUSION

OBJECTIVES

The aims of carrying out this project work are :

# to apply and adapt a variety of problem-solving strategies to

solve problems;

# to improve thinking skills;

# to promote effective mathematical communication;

# to develop mathematical knowledge through problem solving in

a way that increases students’ interest and confidence;

# to use the language of mathematics to express mathematical

ideas precisely;

# to provide learning environment that stimulates and enhances

effective learning;

#to develop positive attitude towards mathematics.

FURTHER EXPLORATION

(a) Linear programming was developed as a discipline in the 1940's,motivated initially by the need to solve complex planning problems in wartime operations. Its development accelerated rapidly in the postwar period as many industries found valuable uses for linear programming. The founders of the subject are generally regarded as George B. Dantzig, who devised the simplex method in 1947, and John von Neumann, who established the theory of duality that same year. The Nobel prize in economics was awarded in 1975 to the mathematician Leonid Kantorovich (USSR) and he economist Tjalling Koopmans (USA) for their contributions to the theory of optimal allocation of resources, in which linear programming played a key role. Many industries use linear programming as a standard tool, e.g. to allocate a finite set of resources in an optimal way. Examples of important application areas include airline crew scheduling, shipping or telecommunication networks, oil refining and blending, and stock and bond portfolio selection.

Linear programming can be applied to various fields of study. It is used in business and economics, but can also be utilized for some engineering problems. Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing.

The simplex method. The simplex method has been the standard technique for solving a linear program since the 1940's. In brief, the simplex method passes from vertex to vertex on the boundary of the feasible polyhedron, repeatedly increasing the objective function until either an optimal solution is found, or it is established that no solution exists. In principle, the time required might be an exponential function of the number of variables, and this can happen in some contrived cases. In practice, however, the method is highly efficient, typically requiring a number of steps which is just a small multiple of the number of variables. Linear programs in thousands or even millions of variables are routinely solved using the simplex method on modern computers.

Efficient, highly sophisticated implementations are available in the form of computer software packages.

Interior-point methods. In 1979, Leonid Khaciyan presented the ellipsoid method, guaranteed to solve any linear program in a number of steps which is a polynomial function of the amount of data defining the linear program. Consequently, the ellipsoid method is faster than the simplex method in contrived cases where the simplex method performs poorly. In practice, however, the simplex method is far superior to the ellipsoid method. In 1984, Narendra Karmarkar introduced an interior-point method for linear programming, combining the desirable theoretical properties of the ellipsoid method and practical advantages of the simplex method. Its success initiated an explosion in the development of interior-point methods. These do not pass from vertex to vertex, but pass only through the interior of the feasible region. Though this property is easy to state, the analysis of interior-point methods is a subtle subject which is much less easily understood than the behaviour of the simplex method. Interior-point methods are now generally considered competitive with the simplex method in most, though not all, applications, and sophisticated software packages implementing them are now available. Whether they will ultimately replace the simplex method in industrial applications is not clear.

An essential component of both the simplex and interior-point methods is the solution of systems of linear equations, which use techniques developed by C.F. Gauss and A. Cholesky in the 18th and 19th centuries. Important generalizations of linear programming include integer programming, quadratic programming, nonlinear programming and stochastic programming. 

Cabinets Cost Area, m2 Volume, m3

X 100 0.6 0.8Y 200 0.8 1.2

i) a) Linear Inequalities

II. 100x + 200y ≤1400

= x+2 y ≤14= y ≤7−12 x

III.0.6 x+0.8 y≤7.2

= y ≤9−34 x

X 0 8 4y 9 3 6

X 0 6

Y 0 9

X 0 8 4Y 7 3 5

ii) Method 1 :

0.8 x+1.2 y=m

0.8 x+1.2 y=12

1.2 y=−0.8 x+12

y=−23x+10

Maximum point (8,3)

0.8(8) + 1.2(3) = 10m3

Method 2 :

Coordinate (4,5)

0.8(4) + 1.2(5) = 9.2m3

Coordinate (8,3)

0.8(8) + 1.2(3) = 10m3

Hence, coordinate (8,3) have a

maximum storage volume.

iii)

X 3 9Y 8 4

Combinations

A b c d e f

Cabinet X 4 5 6 7 8 9

Cabinet Y 5 4 4 3 3 2

Space,m3 9.2 8.8 9.6 9.2 10 9.6

Area, m2 6.4 6.2 6.8 6.6 7.2 7.0

Cost, RM 1400 1300 1400 1300 1400 1300

Combination a (4,5) Combination b (5,4)

Space, 0.8(4)+1.2(5) = 9.2m3 Space, 0.8(5)+1.2(4) = 8.8m3

Area, 0.6(4) + 0.8(5) = 6.4m2 Area, 0.6(5) + 0.8(4) = 6.2m2

Cost, 100(4) + 200(5) = RM1400 Cost, 100(5) + 200(4) = RM1300

Combination c(6,4) Combination d (7,3)

Space, 0.8(6)+1.2(4) = 9.6m3 Space, 0.8(7)+1.2(3) =9.2m3

Area, 0.6(6) + 0.8(4) = 6.8m2 Area, 0.6(7) + 0.8(3) = 6.6m2

Cost, 100(6) + 200(4) = RM1400 Cost, 100(7) + 200(3) = RM1300

Combination e(8,3) Combination f (9,2)

Space, 0.8(8)+1.2(3) = 10m3 Space, 0.8(9)+1.2(2) = 9.6m3

Area, 0.6(8) + 0.8(3) = 7.2m2 Area, 0.6(9) + 0.8(2) = 7.0m2

Cost, 100(8) + 200(3) = RM1400 Cost, 100(9) + 200(2) = RM1300

i˅¿ Based on the table, I choose combination e which is cabinet x = 8 and cabinet y = 3 because it has larger space than others with 10m3 ,larger area than others with 7.2m2 and an allocation of the cost is enough which is RM 1400.

REFLECTION

From this project, I learnt the importance of perseverance as time had been invested to ensure the completion and excellence of this project. Similarly, I learnt the virtue of working together as I have helped and received help from my fellow peers in the production of this project as sharing knowledge is vital in achieving a single goal. Also, I learnt to be thankful and appreciative. This is because, I able to apply my mathematical knowledge in daily life and appreciate the beauty of mathematics. All and all, I have spent countless hours doing this project. I realized that this subject is a compulsory to me. Without it, I can’t fulfill my big dreams and wishes.

It always makes me wonder whyThis subject is so difficult for me…

 I always tried to love every part of it..It always an absolute obstacle for me..

Throughout the day and night… I sacrified precious time to have funAnd even the weekend that I always lookingForward to go out with my family…

 I had spent to do my project..But, when I have completed it..I found that Additional Mathematics is interesting..So, from now on, I will do my best on every second That I will learn Additional Mathematics…

SEKOLAH BERASRAMA PENUH INTEGRASI RAWANG

ADDITIONAL MATHEMATICS’S PROJECT 2015

Name : Aina Zafirah Binti Radzuan

Class :5 Jannatul Khuldi

IC Number : 980126-08-5924Teacher’s Name : Puan Ruslina Binti

Ahmad

SEKOLAH MENENGAH SAINS HULU SELANGOR

Additional Mathematics’s Project 2015

Name: Muhammad Aqishrey Qayyim bin Karmizi

Class : 5 EhsanIC Number : 980429-10-6307

Teacher’s Name :

INTRODUCTION In mathematics, the maximum and minimum 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). 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 objectives of optimization.

In Linear Programming, or sometimes is known as Linear Optimization is a method to achieve the best outcomes (maximum and minimum) in a mathematical model whose requirements are represented by linear

relationships.