Addmaths Project Work 2015

7/23/2019 Addmaths Project Work 2015

1/19

ITION L

M THEM TIC

SPROJECT

WORK

2015

Name: Muhammad Danial Hogan

Class: 5 Science 1

IC: 980508146489

Instructors Name: Mr! "ai#al

1


2/19

Contents

Number Content Page

1 Acknowledgement 3

2 Objective 4

3 Introduction 5

4 Part 1

5 Part 2 !

Part 3 1"

# $urt%er &'(loration 12

! )e*lection 1#

Acknowledgement

2


3/19

First and foremost, I would like to thank Allah for the energy and determination to do this

project. Next, my Additional Mathematics teacher,Mr. Faizal as he gies us consistent guidance

during this project work. !e has "een a ery supportie figure throughout the whole project.

I also would like to gie thanks to all my friends for helping me and always supporting me to

complete this project work. #hey hae done a great jo" at proiding different referenceand

sharing information with other people including me. $ithout them this project would neer hae

had its conclusion.

%ast "ut not least, for their strong support, I would like to express my gratitude to my "eloed

parents. Also for supplying the e&uipments and money needed for the resources to complete this

project. #hey hae always "een "y my side and I hope they will still "e there in the future.

3


4/19

Objectives

#he aims of carrying out this project work are'

I. #o apply and adapt a ariety of pro"lem(soling strategies to sole pro"lems

II. #o improe thinking skills

III. #o promote effectie mathematical communication

IV. #o use the language of mathematics to express mathematical ideas precisely

V.#o proide learning enironment that stimulates and enhances e ectie learning

ff

VI. #o deelop positie attitude towards mathematics

VII. #o deelop mathematical knowledge through pro"lem soling in a way that increases

students interest and con)dence in the su"ject.

4


5/19

Introduction

In mathematics, the maximum and minimum of a function,known collectiely asextrema are the largest and smallest alue that the function takes at a point either within a gien

neigh"ourhood *local or relatie extremum+ or on the function domain in its entirety *glo"al or

a"solute extremum+. ierre de Fermat was one of the first mathematicians to propose a generaltechni&ue *called ade&uality+ for finding maxima and minima. #o locate extreme alues is the

"asic o"jectie of optimization.

%inear programming started when Fermatand %agrangefound calculus("ased formulas

for identifying optima, whileNewtonand-aussproposed iteratie methods for moing towards

an optimum. #he term linear programming for certain optimization cases was due to -eorge /.

0antzig, although much of the theory had "een introduced "y %eonid 1antoroich in 2343.*rogramming in this context does not refer to computer programming, "ut from the use of

program "y the 5nited 6tates military to refer to proposed training and logisticsschedules, which

were the pro"lems 0antzig studied at that time.+ 0antzig pu"lished the 6implex algorithm in2378, and 9ohn on Neumanndeeloped the theory of dualityin the same year.

5
https://en.wikipedia.org/wiki/Pierre_de_Fermathttps://en.wikipedia.org/wiki/Joseph_Louis_Lagrangehttps://en.wikipedia.org/wiki/Isaac_Newtonhttps://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttps://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttps://en.wikipedia.org/wiki/Linear_programminghttps://en.wikipedia.org/wiki/George_Dantzighttps://en.wikipedia.org/wiki/George_Dantzighttps://en.wikipedia.org/wiki/Leonid_Kantorovichhttps://en.wikipedia.org/wiki/Computer_programminghttps://en.wikipedia.org/wiki/Logisticshttps://en.wikipedia.org/wiki/Simplex_algorithmhttps://en.wikipedia.org/wiki/John_von_Neumannhttps://en.wikipedia.org/wiki/Linear_programming#Dualityhttps://en.wikipedia.org/wiki/Joseph_Louis_Lagrangehttps://en.wikipedia.org/wiki/Isaac_Newtonhttps://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttps://en.wikipedia.org/wiki/Linear_programminghttps://en.wikipedia.org/wiki/George_Dantzighttps://en.wikipedia.org/wiki/George_Dantzighttps://en.wikipedia.org/wiki/Leonid_Kantorovichhttps://en.wikipedia.org/wiki/Computer_programminghttps://en.wikipedia.org/wiki/Logisticshttps://en.wikipedia.org/wiki/Simplex_algorithmhttps://en.wikipedia.org/wiki/John_von_Neumannhttps://en.wikipedia.org/wiki/Linear_programming#Dualityhttps://en.wikipedia.org/wiki/Pierre_de_Fermat


6/19

Part 1

*a+

i. Mathematical optimization deals with the pro"lem of finding numerically minimums *ormaximums or zeros+ of a function. In this context, the function is called cost function, or

o"jectie function, or energy.It is a "ranch of mat%ematicsthat focuses on pro"lems

where scarce resources need to "e allocated effectiely, in complex, dynamic and

uncertain conditions. #he program com"ines a solid foundation in math with specialse&uences of courses in economics, "usiness, and management science. In the simplest

case, an optimization pro"lem consists of maximizingor minimizing a real function"y

systematically choosing inputaluesfrom within an allowed set and computingthe alueof the function. #he generalizationof optimization theory and techni&ues to other

formulations comprises a large area of applied mathematics. More generally, optimization

includes finding "est aaila"le alues of some o"jectie function gien a defineddomain*or a set of constraints+, including a ariety of different types of o"jectie

functions and different types of domains.

ii. In mathematical analysis, the maxima and minima *the plural of maximum and

minimum+ of a function, known collectiely as extrema *the plural of extremum+, are the

largest and smallest alue of the function within the entire domain of a function *the

glo"al or a"solute extrema+.

$e say that f*x+ has an a"solute *or glo"al+ maximum at x:c if f*x+ ; f *c+ for eery x in

the domain we are working on. $e say that f*x+ has an a"solute *or glo"al+ minimum at

x:c if if f*x+ < f *c+ for eery 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 collectiely as extrema *the plural of extremum+, are the

largest and smallest alue of the function, within a gien range *the local or relatie

extrema+

$e say that f*x+ has a relatie *or local+ maximum at x:c if f*x+ ; f *c+ for eery x in

some open interal around.

$e say that f*x+ has a relatie *or local+ minimum at x:c if f*x+ < f *c+ for eery x insome open interal around.

6


7/19

0ecide whether you=re going tofind the maximum or minimum

alue. It=s either one or the other,you=re not going to find "oth.

#he maximum > minimum alue ofa &uadratic function occurs at it=s

ertex.For + , a'2-b'-c.

*c("?>7a+ gies the y(alue *or thealue of the function+ at i=s ertex.

If the alue of ais positie,you=re going to get the minimum

alue "ecause as such thepara"ola opens upwards *the

ertex is the lowest the graph canget+ and ice ersa.

Finally, #he alue ofa cannot "e zero.

1ststep

0ifferentiate x withrespect to y.

d+/d' , 2a' - b

2nd

step

0etermine thedifferentiation point aluesin terms of dy>dx. It can "e

found "y setting thesealues e&ual to @ and findthe corresponding alues.

d+/d' , ".2a'-b , "0

1b/2a

3rd

step

6u"stitute this alueof x into y to get theminimum>maximum

points.

*"+ 1stet%od

2nd et%od

7


8/19

Part 2

*a+

4'-2+ , 2""

2'-+ , 1""

+ , 1""2'

Area of the pen : '+

A , '1""2'

A , 1""'2'2

da/d' , 1""4'

$hen the area of the pen is maximum, da/d' , "

1""4' , "

4' , 1""

' , 1""/4

' , 25 m.

Maximum area of the pen :

1"" 2'2

,1""25 2252

,125" m2

8

y

xx xx

y


9/19

*"+

v , %3"2%3"2%

v , %6""12"%-4%2

v , 6""%12"%2-4%3

dv/d% , 6""24"-12%2

$hen olume is maximum, dv/d% , "

12%224"%-6"" , "

%22"%-#5 , "

%5%15 , "

%5 , " or %15 , "

% , 5 % , 15

%must "e lessthan 2

% , 50

v , 6""5 12"52 - 4%3

v , 2""" cm3

9

4@(?h

h

4@


10/19

Part 3

*i+ /ased on the e&uation, a ta"le has "een

constructed where t represents thenum"er of hours starting from @ hours to ?4

hours and represents the num"er of

people.

/ased on the ta"le a"oe, a graph is generated using Microsoft Bxcel application.

10

t/ %ours P/ number o* (eo(le

" "

1 241

2 6""

3 1!""

4 2#""

5 3356

3""

# 3356

! 2#""

6 1!""

1" 6""

11 241

12 "

13 241


11/19

*ii+ #he peak hours with 4C@@ people in the mall is after C hours the mall opens

673" a.m. - %ours , 373" (.m.

*iii+ 8'4@ p.m. is 2@ hours after the malls opens. /ased on the graph, the num"er of people

at the mall at 8'4@ p.m. is 6"" (eo(le.

*i+ /y using the formula,P*t+ :

6

(t)1800cos

D 2E@@

?8@ :

6

(t)1800 cosD 2E@@

6

(t)cos

:25701800

1800

11


12/19

6t=

cos(2(@.7?E

t : 4.E7 hours

t : 4 hours @ minutes

673" a.m. - 3 %ours 5" minutes , 172" (.m.

$urt%er &'(loration

a+ %inear programming is a considera"le field of optimization for seeral reasons. Many

practical pro"lems in operations researchcan "e expressed as linear programming

pro"lems. ertain special cases of linear programming, such as network flowpro"lemsand multicommodity flowpro"lems are considered important enough to hae generated

much research on specialized algorithms for their solution. A num"er of algorithms for

other types of optimization pro"lems work "y soling % pro"lems as su"(pro"lems.

12
https://en.wikipedia.org/wiki/Operations_researchhttps://en.wikipedia.org/wiki/Operations_researchhttps://en.wikipedia.org/wiki/Operations_research


13/19

!istorically, ideas from linear programming hae inspired many of the central concepts

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

of convexityand its generalizations. %ikewise, linear programming is heaily usedinmicroeconomicsand company management, such as planning, production,

transportation, technology and other issues. Although the modern management issues are

eer(changing, most companies would like to maximize profits or minimize costs withlimited resources. #herefore, many issues can "e characterized as linear programming

pro"lems.

#he example of its uses in a daily life includes G rew 6cheduling

i+ Make sure that each flight is coered

ii+ Meet regulations, eg' each pilot can only fly a certain amount each dayiii+ Minimize costs, eg' accommodation for crews staying oernight out of towns,

crews deadheading

i+ $ould like a ro"ust schedule. #he airlines run on small profit margins, so saing a

few percent through good scheduling can make an enormous difference in termsof profita"ility. #hey also use linear programming for yield management

"+ *i+

*a+ I. ost ' 2@@x D ?@@y ; 27@@

II. 6pace ' @.Cx D @.Ey ; 8.?

III. Holume : @.Ex D 2.?y

13
https://en.wikipedia.org/wiki/Microeconomicshttps://en.wikipedia.org/wiki/Microeconomicshttps://en.wikipedia.org/wiki/Microeconomics


14/19

*"+ I.y=

12

x+7

x @ ? 7 C E 2? 27

y 8 C 7 4 2 @

II.y=

34

x+9

x @ ? 7 C E 2@ 2?

y 3 8. C 7. 4 2. @

14


15/19

15


16/19

*ii+ Maximum storage olume

et%od 1 8est using corner (oint o* 9inear Programming :ra(% !0 30 "0 #0 and 120

"

Holume : @.Ex D 2.?y

oordinate 2 ( *E,4+ Holume : @.E*E+ D 2.?*4+ Holume : 2@ cu"ic meter

oordinate ? ( *@,8+ Holume : @.E*@+ D 2.?*8+ Holume : E.7 cu"ic meter

oordinate 4 ( *2?,@+ Holume : @.E*2?+ D 2.?*@+ Holume : 3.C cu"ic meter

#hus the maximum storage olume is 2@ cu"ic meter.

et%od 2 ;sing simultaneous e


17/19

*iii+

ar"inet x ar"inet y #otal ost *M+

7 C 2C@@

2@@

C 7 27@@

8 4 24@@

E 4 27@@

3 ? 24@@

*i+ If I was Aaron, I would choose the com"ination of C ca"inet x and 7 ca"inet y. It

statisfies the term which is the ratio of num"er of ca"inet x to ca"inet y is not less than ?'4.

Furthermore, I hae an allocation of M27@@ for the ca"inets and this com"ination is afforda"leand just perfect for me as it costs exactly M27@@. #he olume of the car"inet is also large

enough.

17


18/19

)e*lection

IJe found a lot of information while conducting this Additional Mathematics project. IJe learnt

the uses of function in our daily lies. I neer thought addmaths could "e used this wayK

#hroughout this project, I hae learned that hardwork is the key to success. My hardships in

doing this project has made me realized that challenges can turn into something "etter. As a

change, I hae came to understand this topic "etter.

I hae also tightened the "onds "etween myself, my instructor and my friends through this

project. $e should always offer help to anyone who asks. I also o"tained a "rand new thinking

skill and can practice effectie mathematical communication thanks to this project work.

In a nutshell, I think this project teaches a lot of moral alues, and also tests the studentsJ

understanding in Additional Mathematics. %et me end this project with a modified lyrics to 9asonMrazJs song, Make it mineL

$ake up eeryone

!ow can you sleep at a time like this

unless youJre gonna score A plus

%isten to his oice,

#he man who teach eerything from just the tip of his thum"s

Making his students understand.

IJm not gonna waste these times

As they are precious

#he success is guaranteed

ItJs just that I know it

18


19/19

IJm gonna make it mine

es IJll make it all mine.

19

Documents

Addmaths Project Work 2015