Content Page

Number Content Page

1 Acknowledgement

2 Objectives

3 Introduction

4 Topic Part I

5 Topic Part II

6 Topic Part III

8 Reflection

OBJECTIVES

At the end of completing this Additional Mathematics Project Work, every form 5 student is hoped to be able to:

Apply and adapt a variety of problem solving strategies to solve routine and non-routine problems.

Experience classroom environments which are challenging, interesting and meaningful hence improve their thinking skills.

Experience classroom environments where knowledge and skills are applied in meaningful ways in solving real-life problems.

Experience classroom environments where expressing ones mathematical thinking, reasoning and communication are highly encouraged and expected.

Experience classroom environments that stimulate and enhance effective learning. Acquire effective mathematical communication through oral and writing, and to use the

language of mathematics to express mathematical ideas correctly and precisely. Enhance acquisition of mathematical knowledge and skills through problem-solving in

ways that increases interest and confidence. Prepare students for the demands of their future undertakings and in workplace. Realize that mathematics is an important and powerful tool in solving real life problems

and hence develop positive attitude towards mathematics. Train themselves not only to be independent learners but also to collaborate, to cooperate,

and to share knowledge in engaging and healthy environment. Use technology especially the ICT appropriately and effectively. Train themselves to appreciate the intrinsic values of mathematics and to become more

creative and innovative. Realize the importance and the beauty of mathematics.

Praised be to Allah for I have fulfilled and achieved the following objectives.

Introduction

The mathematical concept of a function and the name is emerged in the 17th century in

connection with the development of the calculus; for example, the slope dy/dx of a graph at a

point was regarded as a function of the x-coordinate of the point.

Mathematicians of the 18th century typically regarded a function as being defined by an

analytic expression. In the 19th century, the demands of the rigorous development of analysis

by Weierstrass and others, the reformulation of geometry in terms of analysis, and the invention

of set theory by Cantor, eventually led to the much more general modern concept of a function as

a single-valued mapping from one set to another.

Pierre de Fermat was a French lawyer at the Parlement of Toulouse,France, and

a mathematician who is given credit for early developments that led to infinitesimal calculus,

including his technique of adequality. In particular, he is recognized for his discovery of an

original method of finding the greatest and the smallest ordinates of curved lines, which is

analogous to that of the differential calculus, then unknown, and his research into number theory.

He made notable contributions to analytic geometry, probability, and optics. He is best known

for Fermat's Last Theorem, which he described in a note at the margin of a copy

of Diophantus' Arithmetica.

Pierre de Fermat developed the technique of adequality to calculate maxima and

minima of functions, tangents to curves, area, center of mass, least action, and other problems

in mathematical analysis. According to André Weil, Fermat "introduces the technical

termadaequalitas, adaequare etc. Fermat also used his principle to give a mathematical derivation

of Snell's laws of refraction directly from the principle that light takes the quickest path.

There are various types of function. Below are some examples of function

PART I

a) Describe briefly,

Mathematical Optimization is the selection of a best element (with regard to some criteria) from some set of available alternatives. It could also be describe as finding the 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.

Global maximum /minimum or could be defined as global optimum is a selection from a given domain which provides either the highest value (the global maximum) or lowest value (the global minimum), depending on the objective, when a specific function is applied.

Local maximum /minimum can also be expressed as “Relative Maximum”. It is a greatest in a set of points but not the highest when compared to all the values in a set. The set of points can be a global maximum. Refer the diagram below as your reference.

b) Thinking maps are visual learning tools and indeed it helps students, as for myself to memorize better for my studies. And so I have came up with an idea in finding the maximum or minimum value of a quadratic function through the i-Think map.

Example:

PART II

a)

Perimeter: 2x + 2y = 200

Area: xy = A

Make y as subject: y = Ax

Substitute I into

2x + 2 (Ax

) = 200

2x2 + 2A = 200x

Make A as subject;

2A = -2x2 + 200x

A = -x2 + 100x

Differentiate A:

dAdx

= -2x + 100

Maximum area:dAdx

= 0

-2x + 100 = 0

2x = 100

X = 50

y

1

2

3

3 1

x

Maximum area:dAdx

= 0

-2x + 100 = 0

2x = 100

X = 50

Area: A = -x2 + 100x

A = -(50)2 + 100(50)

A = 2500m2

.: The maximum area of the pen is 2500m2

b)

Volume = (30-2h)(30-2h)(h)

V = 900h – 120h2 + 4h3

V = 4h3 – 120h2 + 900h

dvd h

= 12h2 – 240h + 900

Maximum value, dvd h

= 0

12h2 – 240h + 900 = 0

(h – 5) (h – 15) = 0

h = 5, h = 15

30 – 2h

30 – 2h

h

When h = 15

Volume = 4(15)3 -120(15)2 + 900(15)

= 0

When h = 5

Volume = 4(5)3 – 120(5)2 + 900(5)

=2000m3

.: The largest possible volume of the box is 2000m3

PART III

Situation:

A market research company finds that traffic in a local mall over the course of a day could be estimated by the function P (t) = -1800cos ( π/6 t )+1800 where P, is the number of people going to the mall, and t is the time, in hours, after the mall opens. The mall opens at 9.30 a.m.

Questions:

i) Sketch the graph of function P (t).ii) When does the mall reach its peak hours and state the number of people. iii) Estimate the number of people in the mall at 7.30 p.m..iv) Determine the time when the number of people in the mall reaches 2570.

Solutions:

Before proceeding, I have tabulated all of the data from the given function.

Tabulation of data

Time after mall opens

(hours)

Actual time(24 hrs)

People going to the mallP(t)

0 0930 01 1030 2412 1130 9003 1230 18004 1330 27005 1430 33596 1530 36007 1630 33598 1730 27009 1830 180010 1930 90011 2030 24112 2130 0

i) Graph of the number of people going to the mall against time after mall opens

0 2 4 6 8 10 12 140

500

1000

1500

2000

2500

3000

3500

4000

People going to the mall

Time after mall opens (hours)

Peop

le g

oing

to th

e m

all

ii) Based on the data tabulated, the peak hour is at the 6th hour and the number of people is 3600 which recorded the maximum point of the graph.

Peak hours = 3.30 p.m.

The number of people = 3600

iii) Based on the function given, P (t) = -1800cos ( π/6 t)+1800 and the data tabulated, substitute t with 10

P (t) = -1800cos (π/ 6 t) +1800

=-1800cos (180/ 6 (10)) +1800

=900

The number of people in the mall to be estimated at 7.30 p.m. is 900 people

iv) Based on the graph given, the time when the number of people in mall reaches 2570 is at 12.48 p.m.

FURTHER EXPLORATION

Linear Programming (LP)

a) Historical Aspects:

The problem of solving a system of linear inequalities dates back at least as far

as Fourier, who in 1827 published a method for solving them and after whom the method

of Fourier–Motzkin elimination is named.

The first (LP) formulation of a problem that is equivalent to the general linear

programming problem was given by Leonid Kantorovich (pic) 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. About the same time as Kantorovich, the Dutch-American economist T. C.

Koopmans formulated classical economic problems as linear programs. Kantorovich and

Koopmans later shared the 1975 Nobel Prize in economics.  In 1941, Frank Lauren

Hitchcock also formulated transportation problems as linear programs and gave a

solution very similar to the later Simplex method.

The (LP) problem was first shown to be solvable in polynomial time by Leonid

Khachiyan in 1979, but a larger theoretical and practical breakthrough in the field came

in 1984 when Narendra Karmarkar introduced a new interior-point method for solving

(LP) problems.

Necessities in real life situations:

(LP) is a considerable field of optimization for several reasons. Many practical

problems in operations research can be expressed as (LP) problems. Certain special cases

of (LP), such as network flow problems are considered important enough to have

generated much research on specialized algorithms for their solution. A number of

algorithms for other types of optimization problems work by solving (LP) problems as

sub-problems. Historically, ideas from (LP) have inspired many of the central concepts of

optimization theory, such as duality, decomposition, and the importance of convexityand

its generalizations. Likewise, (LP) is heavily used in microeconomics and company

management, such as planning, production, transportation, technology and other issues.

Although the modern management issues are ever-changing, most companies would like

to maximize profits or minimize costs with limited resources. Therefore, many issues can

be characterized as (LP) problems.

Application of (LP)

Agriculture

Assume that a farmer has a piece of farm land, say L km2 is to be planted with either wheat or

barley or some combination of the two. The farmer has a limited amount of fertilizer,

F kilograms, and insecticide, P kilograms. Every square kilometer of wheat requires

F1 kilograms of fertilizer and P1 kilograms of insecticide, while every square kilometer of barley

requires F2 kilograms of fertilizer and P2 kilograms of insecticide. Let S1 be the selling price of

wheat per square kilometer, and S2 be the selling price of barley. If we denote the area of land

planted with wheat and barley by x1 and x2 respectively, then profit can be maximized by

choosing optimal values for x1 and x2. This problem can be expressed with the following linear

programming problem in the standard form:

Maximize: (maximize the revenue—revenue is the "objective

function")

Subject to: (limit on total area)

(limit on fertilizer)

(limit on insecticide)

(cannot plant a negative area)

Printing Industry

A publishing company wishes to set up a printing factory. The company plans to buy two types

of printing machines, model M-10 and N-9. Model M-10 requires 150 m2 of floor area, 4

operation workers and having a printing speed of 600 pages per hour. Model N-9 requires 200 m2

of floor space, 3 operation workers and having a printing speed of 500 pages per hour. The

company is confronted with the following constraints:

I) The factory has a floor area of only 1900 m2.

II) The company will employ not more than 36 operation workers.

III) The number of printing machines must not more than 10 units.

Based on following requirements, the company can use the concept of (LP) in order to help them

with their management. The company holds an operation workers of not more than 36 can be

written as (< 36). Assume x = the number 3 operation workers and having a printing speed of

500 pages per hour, while y = the number 4 operation workers and having a printing speed of

600 pages per hour. The total number of printing machines or (x+y) must not be more than 10

units can be written as (< 10). The factory has a floor area of only 1900 m2. By using the same

concept above, it can be written as (< 1900) and (150y + 200x). The following inequalities can

be reduce into smaller ratio by dividing the number with 50. Those are the results:

I) 3x + 4y < 36

II) x + y < 10

III) 3y + 4x < 38

b) Situation:

Aaron owns a shipping company. He plans to move into his new office which is near to the city

centre. He needs some filing cabinets to organize his files. Cabinet x which costs RM 100 per

unit requires 0.6 square meters of the floor space and can hold 0.8 cubic meters of files. Cabinet

y which cost RM 200 per unit, requires 0.8 square meters of the floor space and can hold 1.2

cubic meters of files. The ratio of the number of cabinet x to the number of cabinet y is not less

than 2:3. Aaron has an allocation of RM 1400 for the cabinets and the office has room for no

more than 7.2 square meters.

Questions:

I) A) Write the inequalities which satisfy all the above constraints.

B) Construct and shade the region that satisfies all the above constraints.

II) Using two different methods, find the maximum storage volume.

III) Aaron plans to buy cabinet x in a range of 4 to 9 units. Tabulate all the possible

combination of the cabinets that he can purchase. Calculate the cost of each combination

IV) If you were Aaron which combination would you choose? Justify your answer and give

your reason.

Solutions:

I) x + 2y < 14

3x + 4y < 36

2x + 3y < 5

II)

REFLECTION

As I look back on what I achieved besides getting to complete this assignment in time, I

had improved on my communicating skills. As I had to deal with the teachers and administrators

for their guidance in improving this project, I learnt to use the right term in communicating with

the elders despite the right intonation to use when interacting.

Moreover, I appreciate mathematics even more the minute after I found the solutions as a

work of art. The concepts or theorems introduced by the mathematicians were actually a

beautiful idea of a piece of art in creating a successful masterpiece, which is the solution of the

given problem. I personally had to brainstorm and do a lot of research regarding the project to

ensure the product that will be presented has its quality. From that, I learnt to be independent as I

did not seek assistance from my elder brother or teachers in every topic I had trouble with.

This is a well-planned strategy made by the government in producing a student self-

access learning method that is strongly applied in the classroom during this era. This is a positive

outcome and I believe by giving this project to the form five students in Malaysia, the

government had achieved their aim which is to ensure the students could apply mathematics

concepts in daily life. Before I end my project work, allow me express my feelings and opinions

through a poem that I made by myself.

Deep below to the world of mathematicsSome might claim it’s pathetic,

But you can learn almost about everything,If you’re into calculating,

You have to master subtraction,Also known to be the brother of addition,You might say ‘Eureka! It is simplified’,

But all your work has to be modified,So now don’t be negative,It’s better to be positive,

Don’t stab yourself with a fork,It’s better to show your work,

Let’s get typical,Use a pencil,I’m not done,

Because Add Math is fun!