ADDITIONAL MATH(1).docx

7/25/2019 ADDITIONAL MATH(1).docx

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SMK PEREMPUAN KAPAR

ADDITIONAL MATHEMATICS

PROJECT WORK 2015

MATHEMATICAL

OPTIMIZATION

NAME : YUvanitha an!"h

CLASS : 5 A#a$n 1

NO%IC : &'1025(10()1&)

TEACHER : *+%TEE TIAN SI


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CONTENTS

TITLE PA,E

ACKNOWLED,E 1

O-JECTI.E 2

INTRODUCTION /()

PART 1 (12

PART 2 12(1&

PART / 20(2/

URTHER EPLORATION 23(2'

RELECTION 2&

CONCLUSION /0


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Acknowledgement

First and foremost, I would like to thank God that finally, I have succeeded in

finishing this project work. I would like to thank my beloved Additional Mathematical

teacher, Mr. ee ian !i for all the guidance he had provided me during the process in

finishing this project work. I also appreciate his patience in guiding me completing this

project work. I would like to give a thousand thanks to my father and mother for giving

me their full support in this project work financially and mentally. hey give me moral

support when I needed it. I would also like to give my thanks to my fellow friends who

had helped me in finding the information that I am clueless of, and the time we spent

together in study groups on finishing this project work. "ast but not least, I would like to

e#press my highest gratitude to all those who give me the possibility to complete this

coursework. I really appreciate all the help I got. Again, thank you very much.


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O-JECTI.Ei% T4 a67 an8 a8at a va+i!t7 49 +46!*

"46vin "t+at!i!" t4 "46v! +46!*"%ii% T4 i*+4v! thin# "#i66%iii% T4 +4*4t! !;!

*ath!*ati


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INTRODUCTION

In *ath!*ati


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Pierre de Fermat

Pierre de Fermat (French: FG89G*a; 17[2]August 1601 12 January 1665 !asaFrench "a!yer at the Parlement#$ %#u"#use&France& and amathematician!h# is gi'en credit

$#r ear"y de'e"#ments that "ed t#in$initesima" ca"cu"us& inc"uding his techni)ue #$ade)ua"ity*+narticu"ar& he is rec#gni,ed $#r his disc#'ery #$ an #rigina" meth#d #$ $inding the greatest and thesma""est#rdinates#$ cur'ed "ines& !hich is ana"#g#us t# that #$ thedi$$erentia" ca"cu"us& thenun-n#!n& and his research int# num.er the#ry* /e made n#ta."e c#ntri.uti#ns t# ana"yticge#metry& r#.a.i"ity& and #tics* /e is .est -n#!n $#r Fermats ast %he#rem& !hich hedescri.ed in a n#te at the margin #$ a c#y #$ i#hantusArithmetica*

Life and work

Fermat !as .#rn in the $irst decade #$ the 17th century in3eaum#nt4de4#magne(resent4day %arn4et4ar#nne& France; the "ate 15th4century mansi#n !here Fermat !as .#rn is n#! a

museum* /e !as $r#m asc#ny& !here his $ather& #mini)ue Fermat& !as a !ea"thy "eathermerchant& and ser'ed three #ne4year terms as #ne #$ the $#ur c#nsu"s #$ 3eaum#nt4de4#magne*/is m#ther !as either Fran#ise a,eneu'e #r "aire de #ng* Pierre had a .r#ther and t!#sisters and !as a"m#st certain"y .r#ught u in the t#!n #$ his .irth* %here is "itt"e e'idencec#ncerning his sch##" educati#n& .ut it !as r#.a."y at the #""8ge de 9a'arrein #ntau.an*

/e attended the ni'ersity #$


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E#r-

Fermats i#neering !#r- in ana"ytic ge#metry !as circu"ated in manuscrit $#rm in 16>6&redating the u."icati#n #$ escartes $am#us La gomtrie*%his manuscrit !as u."ished#sthum#us"y in 167@ in Caria #era mathematica& asAd Locos PlanosetSolidosIsagoge&(+ntr#ducti#n t# P"ane and D#"id #ci* [5]

+n Methodusaddisquirendammaximamet minimaand in De tangentibuslinearumcurvarum& Fermatde'e"#ed a meth#d (ade)ua"ity $#r determining ma?ima& minima& and tangents t# 'ari#uscur'es that !as e)ui'a"ent t# di$$erentia" ca"cu"us*[6][7]+n these !#r-s& Fermat #.tained a techni)ue$#r $inding the centers #$ gra'ity #$ 'ari#us "ane and s#"id $igures& !hich "ed t# his $urther !#r-in)uadrature*

Pierre de Fermat

Fermat !as the $irst ers#n -n#!n t# ha'e e'a"uated the integra" #$ genera" #!er $uncti#ns*

sing an ingeni#us tric-& he !as a."e t# reduce this e'a"uati#n t# the sum #$ge#metric series*[H]%he resu"ting $#rmu"a !as he"$u" t# 9e!t#n& and then ei.ni,& !hen they indeendent"yde'e"#ed the$undamenta" the#rem #$ ca"cu"us*[citation needed]

+n num.er the#ry& Fermat studied Pe""s e)uati#n& er$ect num.ers&amica."e num.ersand !hat!#u"d "ater .ec#meFermat num.ers* +t !as !hi"e researching er$ect num.ers that hedisc#'ered the "itt"e the#rem* /e in'ented a $act#ri,ati#n meth#dIFermats $act#ri,ati#n meth#dIas !e"" as the r##$ techni)ue #$ in$inite descent& !hich he used t# r#'e Fermats right triang"ethe#rem!hich inc"udes as a c#r#""ary Fermats ast %he#rem $#r the case n G* Fermatde'e"#ed thet!#4s)uare the#rem&and the #"yg#na" num.er the#rem& !hich states that eachnum.er is a sum #$ threetriangu"ar num.ers&$#ur s)uare num.ers& $i'eentag#na" num.ers& and

s# #n*

A"th#ugh Fermat c"aimed t# ha'e r#'ed a"" his arithmetic the#rems& $e! rec#rds #$ his r##$sha'e sur'i'ed* any mathematicians& inc"uding auss&d#u.ted se'era" #$ his c"aims& esecia""ygi'en the di$$icu"ty #$ s#me #$ the r#."ems and the "imited mathematica" meth#ds a'ai"a."e t#Fermat* /is $am#us ast %he#rem!as $irst disc#'ered .y his s#n in the margin #n his $athersc#y #$ an editi#n #$ i#hantus& and inc"uded the statement that the margin !as t## sma"" t#inc"ude the r##$* /e had n#t .#thered t# in$#rm e'enarin ersenne#$ it* +t !as n#t r#'edunti" 1@@G .yDir Andre! Ei"es& using techni)ues una'ai"a."e t# Fermat*
https://en.wikipedia.org/wiki/La_G%C3%A9om%C3%A9triehttps://en.wikipedia.org/wiki/La_G%C3%A9om%C3%A9triehttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-5https://en.wikipedia.org/wiki/Adequalityhttps://en.wikipedia.org/wiki/Differential_calculushttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-Pellegrino-6https://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-Pellegrino-6https://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-7https://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-7https://en.wikipedia.org/wiki/Numerical_integrationhttps://en.wikipedia.org/wiki/Numerical_integrationhttps://en.wikipedia.org/wiki/Geometric_serieshttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-quadrature-8https://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-quadrature-8https://en.wikipedia.org/wiki/Isaac_Newtonhttps://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibnizhttps://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttps://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttps://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttps://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttps://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttps://en.wikipedia.org/wiki/Pell's_equationhttps://en.wikipedia.org/wiki/Perfect_numberhttps://en.wikipedia.org/wiki/Perfect_numberhttps://en.wikipedia.org/wiki/Amicable_numberhttps://en.wikipedia.org/wiki/Fermat_numbershttps://en.wikipedia.org/wiki/Fermat_numbershttps://en.wikipedia.org/wiki/Fermat's_little_theoremhttps://en.wikipedia.org/wiki/Fermat's_factorization_methodhttps://en.wikipedia.org/wiki/Infinite_descenthttps://en.wikipedia.org/wiki/Fermat's_right_triangle_theoremhttps://en.wikipedia.org/wiki/Fermat's_right_triangle_theoremhttps://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squareshttps://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squareshttps://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squareshttps://en.wikipedia.org/wiki/Fermat_polygonal_number_theoremhttps://en.wikipedia.org/wiki/Triangular_numberhttps://en.wikipedia.org/wiki/Triangular_numberhttps://en.wikipedia.org/wiki/Lagrange's_four-square_theoremhttps://en.wikipedia.org/wiki/Lagrange's_four-square_theoremhttps://en.wikipedia.org/wiki/Pentagonal_numberhttps://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttps://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttps://en.wikipedia.org/wiki/Fermat's_Last_Theoremhttps://en.wikipedia.org/wiki/Marin_Mersennehttps://en.wikipedia.org/wiki/Wiles's_proof_of_Fermat's_Last_Theoremhttps://en.wikipedia.org/wiki/Wiles's_proof_of_Fermat's_Last_Theoremhttps://en.wikipedia.org/wiki/La_G%C3%A9om%C3%A9triehttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-5https://en.wikipedia.org/wiki/Adequalityhttps://en.wikipedia.org/wiki/Differential_calculushttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-Pellegrino-6https://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-7https://en.wikipedia.org/wiki/Numerical_integrationhttps://en.wikipedia.org/wiki/Geometric_serieshttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-quadrature-8https://en.wikipedia.org/wiki/Isaac_Newtonhttps://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibnizhttps://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttps://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttps://en.wikipedia.org/wiki/Pell's_equationhttps://en.wikipedia.org/wiki/Perfect_numberhttps://en.wikipedia.org/wiki/Amicable_numberhttps://en.wikipedia.org/wiki/Fermat_numbershttps://en.wikipedia.org/wiki/Fermat's_little_theoremhttps://en.wikipedia.org/wiki/Fermat's_factorization_methodhttps://en.wikipedia.org/wiki/Infinite_descenthttps://en.wikipedia.org/wiki/Fermat's_right_triangle_theoremhttps://en.wikipedia.org/wiki/Fermat's_right_triangle_theoremhttps://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squareshttps://en.wikipedia.org/wiki/Fermat_polygonal_number_theoremhttps://en.wikipedia.org/wiki/Triangular_numberhttps://en.wikipedia.org/wiki/Lagrange's_four-square_theoremhttps://en.wikipedia.org/wiki/Pentagonal_numberhttps://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttps://en.wikipedia.org/wiki/Fermat's_Last_Theoremhttps://en.wikipedia.org/wiki/Marin_Mersennehttps://en.wikipedia.org/wiki/Wiles's_proof_of_Fermat's_Last_Theorem


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A"th#ugh he care$u""y studied& and dre! insirati#n $r#m i#hantus& Fermat .egan a di$$erenttraditi#n* i#hantus !as c#ntent t# $ind a sing"e s#"uti#n t# his e)uati#ns& e'en i$ it !ere anundesired $racti#na" #ne* Fermat !as interested #n"y in integer s#"uti#ns t# his i#hantinee)uati#ns&and he "##-ed $#r a"" #ssi."e genera" s#"uti#ns* /e #$ten r#'ed that certain e)uati#nshadn# s#"uti#n& !hich usua""y .a$$"ed his c#ntem#raries*[citation needed]

%hr#ugh their c#rres#ndence in 165G& Fermat and 3"aise Pasca"he"ed "ay the $undamenta"gr#und!#r- $#r the the#ry #$ r#.a.i"ity* Fr#m this .rie$ .ut r#ducti'e c#""a.#rati#n #nthe r#."em #$ #ints&they are n#! regarded as K#int $#unders #$r#.a.i"ity the#ry*[@]Fermat iscredited !ith carrying #ut the $irst e'er rig#r#us r#.a.i"ity ca"cu"ati#n* +n it& he !as as-ed .y ar#$essi#na" gam."er!hy i$ he .et #n r#""ing at "east #ne si? in $#ur thr#!s #$ a die he !#n in the"#ng term& !hereas .etting #n thr#!ing at "east #ne d#u."e4si? in 2G thr#!s #$ t!# dice resu"ted inhis "#sing* Fermat su.se)uent"y r#'ed !hy this !as the case mathematica""y*[10]

Fermatsrinci"e #$ "east time (!hich he used t# deri'e Dne""s "a! in 1657 !as the$irst 'ariati#na" rinci"e[11]enunciated in hysics since /er# #$ A"e?andria descri.ed a rinci"e #$"east distance in the $irst century B* +n this !ay& Fermat is rec#gni,ed as a -ey $igure in the

hist#rica" de'e"#ment #$ the $undamenta" rinci"e #$ "east acti#nin hysics* %he termsFermatsrinci"eand Fermat functional!ere named in rec#gniti#n #$ this r#"e*[12]

Assessment of his work

/#"#grahic !i""hand!ritten .y Fermat #n G arch 1660 I -et at the eartmenta" Archi'es

#$/aute4ar#nne&in%#u"#use

%#gether !ith Len= escartes& Fermat !as #ne #$ the t!# "eading mathematicians #$ the $irst ha"$#$ the 17th century* Acc#rding t# Peter * 3ernstein& in his .##-Against the Gods& Fermat !as amathematician #$ rare #!er* /e !as an indeendent in'ent#r #$ ana"ytic ge#metry& hec#ntri.uted t# the ear"y de'e"#ment #$ ca"cu"us& he did research #n the !eight #$ the earth& and

he !#r-ed #n "ight re$racti#n and #tics* +n the c#urse #$ !hat turned #ut t# .e an e?tendedc#rres#ndence !ith Pasca"& he made a signi$icant c#ntri.uti#n t# the the#ry #$ r#.a.i"ity* 3utFermats cr#!ning achie'ement !as in the the#ry #$ num.ers*[1G]

Legarding Fermats !#r- in ana"ysis&+saac 9e!t#n!r#te that his #!n ear"y ideas a.#ut ca"cu"uscame direct"y $r#m Fermats !ay #$ dra!ing tangents*[15]
https://en.wikipedia.org/wiki/Diophantine_equationhttps://en.wikipedia.org/wiki/Diophantine_equationhttps://en.wikipedia.org/wiki/Diophantine_equationhttps://en.wikipedia.org/wiki/Empty_sethttps://en.wikipedia.org/wiki/Empty_sethttps://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttps://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttps://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttps://en.wikipedia.org/wiki/Blaise_Pascalhttps://en.wikipedia.org/wiki/Problem_of_pointshttps://en.wikipedia.org/wiki/Problem_of_pointshttps://en.wikipedia.org/wiki/Probability_theoryhttps://en.wikipedia.org/wiki/Probability_theoryhttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-mactutor-9https://en.wikipedia.org/wiki/Gamblerhttps://en.wikipedia.org/wiki/Dicehttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-10https://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-10https://en.wikipedia.org/wiki/Principle_of_least_timehttps://en.wikipedia.org/wiki/Snell's_lawhttps://en.wikipedia.org/wiki/History_of_variational_principles_in_physicshttps://en.wikipedia.org/wiki/History_of_variational_principles_in_physicshttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-variational-11https://en.wikipedia.org/wiki/Hero_of_Alexandriahttps://en.wikipedia.org/wiki/Principle_of_least_actionhttps://en.wikipedia.org/wiki/Fermat's_principlehttps://en.wikipedia.org/wiki/Fermat's_principlehttps://en.wikipedia.org/wiki/Fermat's_principlehttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-functional-12https://en.wikipedia.org/wiki/Holographic_willhttps://en.wikipedia.org/wiki/Haute-Garonnehttps://en.wikipedia.org/wiki/Haute-Garonnehttps://en.wikipedia.org/wiki/Toulousehttps://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-Bernstein-14https://en.wikipedia.org/wiki/Isaac_Newtonhttps://en.wikipedia.org/wiki/Isaac_Newtonhttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-Simmons-15https://en.wikipedia.org/wiki/Diophantine_equationhttps://en.wikipedia.org/wiki/Diophantine_equationhttps://en.wikipedia.org/wiki/Empty_sethttps://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttps://en.wikipedia.org/wiki/Blaise_Pascalhttps://en.wikipedia.org/wiki/Problem_of_pointshttps://en.wikipedia.org/wiki/Probability_theoryhttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-mactutor-9https://en.wikipedia.org/wiki/Gamblerhttps://en.wikipedia.org/wiki/Dicehttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-10https://en.wikipedia.org/wiki/Principle_of_least_timehttps://en.wikipedia.org/wiki/Snell's_lawhttps://en.wikipedia.org/wiki/History_of_variational_principles_in_physicshttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-variational-11https://en.wikipedia.org/wiki/Hero_of_Alexandriahttps://en.wikipedia.org/wiki/Principle_of_least_actionhttps://en.wikipedia.org/wiki/Fermat's_principlehttps://en.wikipedia.org/wiki/Fermat's_principlehttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-functional-12https://en.wikipedia.org/wiki/Holographic_willhttps://en.wikipedia.org/wiki/Haute-Garonnehttps://en.wikipedia.org/wiki/Toulousehttps://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-Bernstein-14https://en.wikipedia.org/wiki/Isaac_Newtonhttps://en.wikipedia.org/wiki/Pierre_de_Fermat#cite_note-Simmons-15


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the'a"ue#$ the $uncti#n* %he genera"i,ati#n #$ #timi,ati#n the#ry and techni)ues t# #ther

$#rmu"ati#ns c#mrises a "arge area #$a"ied mathematics* #re genera""y& #timi,ati#n

inc"udes $inding .est a'ai"a."e 'a"ues #$ s#me #.Kecti'e $uncti#n gi'en a de$ined d#main(#r

a set #$ c#nstraints& inc"uding a 'ariety #$ di$$erent tyes #$ #.Kecti'e $uncti#ns and di$$erent

tyes #$ d#mains

Graph of aparaboloidgiven by f(x, y$ ) *(x+ y+$ -. he globalma#imumat (, , -$ is

indicated by a red dot.

Global Ma#imum/Minimum

In mathematical analysis,the ma#ima and minima (the plural of ma#imum and

minimum$ of a function, known collectively as e#trema (the plural of e#tremum$, are the

largest and smallest value of the function, either within a given range (the

local or relative e#trema$ or on the entire domain of a function (the global or

absolute e#trema$. 0ierre de Fermatwas one of the first mathematicians to propose a

general techni1ue, ade1uality, for finding the ma#ima and minima of functions.

As defined in set theory, the ma#imum and minimum of a setare the greatest and least

elementsin the set, respectively. 2nbounded infinite sets, such as the set of real numbers,

have no minimum or ma#imum.
https://en.wikipedia.org/wiki/Value_(mathematics)https://en.wikipedia.org/wiki/Value_(mathematics)https://en.wikipedia.org/wiki/Applied_mathematicshttps://en.wikipedia.org/wiki/Applied_mathematicshttps://en.wikipedia.org/wiki/Domain_of_a_functionhttps://en.wikipedia.org/wiki/Paraboloidhttps://en.wikipedia.org/wiki/Maximum_(mathematics)https://en.wikipedia.org/wiki/Mathematical_analysishttps://en.wikipedia.org/wiki/Mathematical_analysishttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Domain_of_a_functionhttps://en.wikipedia.org/wiki/Pierre_de_Fermathttps://en.wikipedia.org/wiki/Adequalityhttps://en.wikipedia.org/wiki/Set_theoryhttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Greatest_elementhttps://en.wikipedia.org/wiki/Greatest_elementhttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Value_(mathematics)https://en.wikipedia.org/wiki/Applied_mathematicshttps://en.wikipedia.org/wiki/Domain_of_a_functionhttps://en.wikipedia.org/wiki/Paraboloidhttps://en.wikipedia.org/wiki/Maximum_(mathematics)https://en.wikipedia.org/wiki/Mathematical_analysishttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Domain_of_a_functionhttps://en.wikipedia.org/wiki/Pierre_de_Fermathttps://en.wikipedia.org/wiki/Adequalityhttps://en.wikipedia.org/wiki/Set_theoryhttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Greatest_elementhttps://en.wikipedia.org/wiki/Greatest_elementhttps://en.wikipedia.org/wiki/Real_number


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"ocal and global ma#ima and minima for cos(34x$/x, .56 x65.5

"ocal Ma#imum/Minimum

Function can have hill and valleys:places where they reach a minimum or ma#imum

value. It may not be the minimum or ma#imum for the whole function, but locally it is.


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"ocal Ma#imum

First, we need to choose an interval7

Then we can say that a local maximumis the point where:

Th! h!iht 49 th! 9$n


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

"ocal Minimum

"ikewise, a local minimumis7

f(a$ 6 f(#$ for all # in the interval

he plural of Ma#imum is Maxima

he plural of Minimum is Minima

Ma#ima and Minima are collectively called Extrema

Global (or Absolute$ Ma#imum and Minimum

he ma#imum or minimum over the entire functionis called an 9Absolute9 or 9Global9

ma#imum or minimum.

here is only one global ma#imum (and one global minimum$ but there can be more than

one local ma#imum or minimum.

Assumingthis function continues downwards to left or right7


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he Global Ma#imum is about 3.:

he Global Minimum is *Infinity

Pa+t 2

aEnH!n


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Method 5 (%ifferentiation$

-# ;y) ; -# y ) 5 > ;(;=$

y ) 5 > ;# ) =

Area of the pen ) #y

A ) # (5 > ;#$ Ma#imum area of the pen

) 5#? ;#; ) =m #;=m

dAdx ) 5 > -# ) 5;=m

;

dA

dx=

, ) 5 > -#

-# ) 5

# ) ;=

Method ; (@ompleting he !1uare$

A ) 5# > ;#;

) ?;#; 5#

) ?;(#; ? =#$

) ?;(# ?;=$;> B;=C

) ?;(# > ;=$;5;=


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!ince aD, f(#$ has a ma#imum value. he ma#imum value of A is 5;=. A is ma#imum

when # > ;=),that is #);=.

#y)5;=

;=y ) 5;=

y ) =

Ma#imum value of the pen is 5;=m;

Method 3 (Graph of ma#imum and minimum solve$

-# ;y ) ;

;y ) ; > -#

y ) 5 > ;#

A ) #y

) #(5 > ;#$


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) 5# > ;#;

# = 5 5= ; ;= 3 3= - -= =

y E : B = - 3 ; 5

A -= 5= 5; 5;= 5; 5= -=

0 10 20 /0 30 50 )0

0

200

300

)00

'00

1000

1200

1300

Length(m)

Area (m2)

b$e'a is helping Hn !hah to make a bo# without the top. he bo# is made by cutting

away four s1uares from the corners of a 3cm piece of cupboard as shown in Figure 5

and bending up the resulting cupboard to form the walls of the bo#.h

i$+! 1

/0 2h


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Find the largest possible volume of the bo#.

Method 5 (%ifferentiation$

) (3 > ;h$(3 > ;h$h

) (E > 5;h -h;$h

) Eh > 5;h; -h3

dV

d h ) 5;h;> ;-h E

dV

d h ) , ) 5;h;> ;-h E

) (h > =$(h > 5=$

h) =, 5=

5;(=$;E(=$

) ;cm3 (=, ;$

5;(5=$;E(5=$

), rejected


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dV

d h );-h?;-

(=, ;$d

2V

dh

2 ) ;-(=$ > ;-

) ?5; ,J

(=, ;$ is a ma#imum point,

Ma#imum volume of the bo# ) -(=$3> 5;(=$;E(=$

) ;cm3 (=, ;$

Method ;(Graph$

) (3 > ;h$(3 > ;h$ h

) (E > 5;h -h;$ h


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) Eh > 5;h; -h3

h 5 ; 3 - = B : E 5 55 5; 53 5

-

:

-

53=

;

5:;

5E3

B

;

5E-

-

5:E

;

5=B

5;E

B

5

:

-

-3

;

;

=

B

0 2 3 ) ' 10 12 13 1)

0

500

1000

1500

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A market research company finds that traffic in a local mall over the course of a day

could be estimated by the functionP (t)=1800cos(6 t)+1800 where 0, is the

number of people going to the mall, and t is the time, in hours, after the mall opens. he

mall opens at E.3am.

i. !ketch the graph of function 0(t$

ii.


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t(time)

(t)

ii. he mall reaches its peak hours B hours after opening at E.3am.,which is at

3.3pm. he number of people are 3B.

iii. :.3 ) 5

Kased on the graph,when t ) 5, p(t$ ) E

he number of people in the mall at :.3pm.is E people.

iv. Kased on the graph,

when p(t$ is ;=:, t )3.E , .5

3.E ) 3 hours =- minutes

&%/0*% / h+" 53 *in$t!"Q1%23*

'%1 Q ' h4$+" ) *in$t!"

&%/0*% 'h+" ) *in$t!" Q 5%/)*%

Th! *a66 +!a


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A!i" Title


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$th!+ E@64+ati4n

b$ Aaron owns a shipping company. Le plans to move into his new office which is near to

the city centre. Le needs some filing cabinets to organi'e his file. @abinets # which costM5 per unit, re1uires .B s1uare meters of the floor space and can hold . cubic

meters of files. @abinet y which cost M ; per unit, re1uires . s1uare meters of the

floor space and can hold 5.; cubic meters of files. he ratio of the number of cabinet # to

the number of cabinet y is not less than ;73. Aaron has an allocation of M 5- for the

cabinets and the office has room for no more than :.; s1uare meters.

i. 2sing the given information,

(a$


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Linear programming(LP; a"s# ca""ed linear optimization is a meth#d t# achie'e the .est

#utc#me (such as ma?imum r#$it #r "#!est c#st in a mathematica" m#de"!h#se

re)uirements are reresented .y "inear re"ati#nshis* inear r#gramming is a secia" case

#$ mathematica" r#gramming (mathematica" #timi,ati#n*

#re $#rma""y& "inear r#gramming is a techni)ue $#r the#timi,ati#n#$ a "inear#.Kecti'e

$uncti#n& su.Kect t#"inear e)ua"ity and "inear ine)ua"ityc#nstraints*+ts$easi."e regi#nis

a c#n'e? #"yt#e& !hich is a set de$ined as the intersecti#n#$ $inite"y many ha"$ saces&

each #$ !hich is de$ined .y a "inear ine)ua"ity* +ts #.Kecti'e $uncti#n is a rea"4'a"ued a$$ine

$uncti#nde$ined #n this #"yhedr#n* A "inear r#gramming a"g#rithm$inds a #int in the

#"yhedr#n !here this $uncti#n has the sma""est (#r "argest 'a"ue i$ such a #int e?ists*

inear r#grams are r#."ems that can .e e?ressed in can#nica" $#rm:

!here ? reresents the 'ect#r #$ 'aria."es (t# .e determined& c and . are 'ect#rs#$

(-n#!n c#e$$icients&Ais a (-n#!n matri?#$ c#e$$icients& and is the matri?

trans#se*%he e?ressi#n t# .e ma?imi,ed #r minimi,ed is ca""ed the ob*ective

function(c%? in this case* %he ine)ua"itiesA? M . and ? N 0 are the c#nstraints !hich

seci$y ac#n'e? #"yt#e#'er !hich the #.Kecti'e $uncti#n is t# .e #timi,ed* +n thisc#nte?t& t!# 'ect#rs are c#mara."e!hen they ha'e the same dimensi#ns* +$ e'ery

entry in the $irst is "ess4than #r e)ua"4t# the c#rres#nding entry in the sec#nd then !e

can say the $irst 'ect#r is "ess4than #r e)ua"4t# the sec#nd 'ect#r*

inear r#gramming can .e a"ied t# 'ari#us $ie"ds #$ study* +t is used in .usiness

and ec#n#mics& .ut can a"s# .e uti"i,ed $#r s#me engineering r#."ems* +ndustries that

use "inear r#gramming m#de"s inc"ude trans#rtati#n& energy& te"ec#mmunicati#ns& and

manu$acturing* +t has r#'ed use$u" in m#de"ing di'erse tyes #$ r#."ems in

"anning& r#uting& schedu"ing& assignment&and design*
https://en.wikipedia.org/wiki/Mathematical_modelhttps://en.wikipedia.org/wiki/Mathematical_optimizationhttps://en.wikipedia.org/wiki/Mathematical_optimizationhttps://en.wikipedia.org/wiki/Linearhttps://en.wikipedia.org/wiki/Objective_functionhttps://en.wikipedia.org/wiki/Objective_functionhttps://en.wikipedia.org/wiki/Linear_equalityhttps://en.wikipedia.org/wiki/Linear_inequalityhttps://en.wikipedia.org/wiki/Constraint_(mathematics)https://en.wikipedia.org/wiki/Constraint_(mathematics)https://en.wikipedia.org/wiki/Constraint_(mathematics)https://en.wikipedia.org/wiki/Feasible_regionhttps://en.wikipedia.org/wiki/Convex_polytopehttps://en.wikipedia.org/wiki/Intersection_(mathematics)https://en.wikipedia.org/wiki/Half-space_(geometry)https://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Affine_functionhttps://en.wikipedia.org/wiki/Affine_functionhttps://en.wikipedia.org/wiki/Algorithmhttps://en.wikipedia.org/wiki/Canonical_formhttps://en.wikipedia.org/wiki/Canonical_formhttps://en.wikipedia.org/wiki/Vector_spacehttps://en.wikipedia.org/wiki/Matrix_(mathematics)https://en.wikipedia.org/wiki/Matrix_transposehttps://en.wikipedia.org/wiki/Matrix_transposehttps://en.wikipedia.org/wiki/Matrix_transposehttps://en.wikipedia.org/wiki/Convex_polytopehttps://en.wikipedia.org/wiki/Comparabilityhttps://en.wikipedia.org/wiki/Economicshttps://en.wikipedia.org/wiki/Routinghttps://en.wikipedia.org/wiki/Scheduling_(production_processes)https://en.wikipedia.org/wiki/Assignment_problemhttps://en.wikipedia.org/wiki/Assignment_problemhttps://en.wikipedia.org/wiki/Mathematical_modelhttps://en.wikipedia.org/wiki/Mathematical_optimizationhttps://en.wikipedia.org/wiki/Mathematical_optimizationhttps://en.wikipedia.org/wiki/Linearhttps://en.wikipedia.org/wiki/Objective_functionhttps://en.wikipedia.org/wiki/Objective_functionhttps://en.wikipedia.org/wiki/Linear_equalityhttps://en.wikipedia.org/wiki/Linear_inequalityhttps://en.wikipedia.org/wiki/Constraint_(mathematics)https://en.wikipedia.org/wiki/Feasible_regionhttps://en.wikipedia.org/wiki/Convex_polytopehttps://en.wikipedia.org/wiki/Intersection_(mathematics)https://en.wikipedia.org/wiki/Half-space_(geometry)https://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Affine_functionhttps://en.wikipedia.org/wiki/Affine_functionhttps://en.wikipedia.org/wiki/Algorithmhttps://en.wikipedia.org/wiki/Canonical_formhttps://en.wikipedia.org/wiki/Vector_spacehttps://en.wikipedia.org/wiki/Matrix_(mathematics)https://en.wikipedia.org/wiki/Matrix_transposehttps://en.wikipedia.org/wiki/Matrix_transposehttps://en.wikipedia.org/wiki/Convex_polytopehttps://en.wikipedia.org/wiki/Comparabilityhttps://en.wikipedia.org/wiki/Economicshttps://en.wikipedia.org/wiki/Routinghttps://en.wikipedia.org/wiki/Scheduling_(production_processes)https://en.wikipedia.org/wiki/Assignment_problem


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History

e#nid Oant#r#'ich

%he r#."em #$ s#"'ing a system #$ "inear ine)ua"ities dates .ac- at "east as $ar

asF#urier& !h# in 1H27 u."ished a meth#d $#r s#"'ing them&[1]and a$ter !h#m the

meth#d #$ F#urier#t,-in e"iminati#n is named*

%he $irst "inear r#gramming $#rmu"ati#n #$ a r#."em that is e)ui'a"ent t# the genera"

"inear r#gramming r#."em !as gi'en .y e#nid Oant#r#'ichin 1@>@& !h# a"s#

r##sed a meth#d $#r s#"'ing it*[2]/e de'e"#ed it during E#r"d Ear ++as a !ay t# "an

e?enditures and returns s# as t# reduce c#sts t# the army and increase "#sses incurred

.y the enemy* A.#ut the same time as Oant#r#'ich& the utch4American ec#n#mist %* *

O##mans $#rmu"ated c"assica" ec#n#mic r#."ems as "inear r#grams* Oant#r#'ich and

O##mans "ater shared the 1@759#.e" ri,e in ec#n#mics*[1]+n 1@G1& Fran- auren

/itchc#c-a"s# $#rmu"ated trans#rtati#n r#."ems as "inear r#grams and ga'e a

s#"uti#n 'ery simi"ar t# the "ater Dim"e? meth#d;[2]/itchc#c- had died in 1@57 and the

9#.e" ri,e is n#t a!arded #sthum#us"y*

uring 1@G641@G7& e#rge 3* ant,igindeendent"y de'e"#ed genera" "inear

r#gramming $#rmu"ati#n t# use $#r "anning r#."ems in D Air F#rce* +n 1@G7& ant,ig

a"s# in'ented thesim"e? meth#dthat $#r the $irst time e$$icient"y tac-"ed the "inear

r#gramming r#."em in m#st cases* Ehen ant,ig arranged meeting !ithJ#hn '#n

9eumannt# discuss his Dim"e? meth#d& 9eumann immediate"y c#nKectured the the#ry

#$ dua"ity .y rea"i,ing that the r#."em he had .een !#r-ing ingame the#ry!as

e)ui'a"ent* ant,ig r#'ided $#rma" r##$ in an unu."ished re#rt A %he#rem #n inear

+ne)ua"ities #n January 5& 1@GH* [>]P#st!ar& many industries $#und its use in their dai"y

"anning*
https://en.wikipedia.org/wiki/Leonid_Kantorovichhttps://en.wikipedia.org/wiki/Joseph_Fourierhttps://en.wikipedia.org/wiki/Linear_programming#cite_note-Sierksma2001-1https://en.wikipedia.org/wiki/Fourier%E2%80%93Motzkin_eliminationhttps://en.wikipedia.org/wiki/Leonid_Kantorovichhttps://en.wikipedia.org/wiki/Linear_programming#cite_note-Schrijver1998-2https://en.wikipedia.org/wiki/Linear_programming#cite_note-Schrijver1998-2https://en.wikipedia.org/wiki/World_War_IIhttps://en.wikipedia.org/wiki/Tjalling_Koopmanshttps://en.wikipedia.org/wiki/Tjalling_Koopmanshttps://en.wikipedia.org/wiki/Nobel_prize_in_economicshttps://en.wikipedia.org/wiki/Nobel_prize_in_economicshttps://en.wikipedia.org/wiki/Linear_programming#cite_note-Sierksma2001-1https://en.wikipedia.org/wiki/Linear_programming#cite_note-Sierksma2001-1https://en.wikipedia.org/wiki/Frank_Lauren_Hitchcockhttps://en.wikipedia.org/wiki/Frank_Lauren_Hitchcockhttps://en.wikipedia.org/wiki/Linear_programming#cite_note-Schrijver1998-2https://en.wikipedia.org/wiki/George_Dantzighttps://en.wikipedia.org/wiki/Simplex_algorithmhttps://en.wikipedia.org/wiki/Simplex_algorithmhttps://en.wikipedia.org/wiki/John_von_Neumannhttps://en.wikipedia.org/wiki/John_von_Neumannhttps://en.wikipedia.org/wiki/Linear_programming#Dualityhttps://en.wikipedia.org/wiki/Game_theoryhttps://en.wikipedia.org/wiki/Linear_programming#cite_note-3https://en.wikipedia.org/wiki/Linear_programming#cite_note-3https://en.wikipedia.org/wiki/Leonid_Kantorovichhttps://en.wikipedia.org/wiki/Joseph_Fourierhttps://en.wikipedia.org/wiki/Linear_programming#cite_note-Sierksma2001-1https://en.wikipedia.org/wiki/Fourier%E2%80%93Motzkin_eliminationhttps://en.wikipedia.org/wiki/Leonid_Kantorovichhttps://en.wikipedia.org/wiki/Linear_programming#cite_note-Schrijver1998-2https://en.wikipedia.org/wiki/World_War_IIhttps://en.wikipedia.org/wiki/Tjalling_Koopmanshttps://en.wikipedia.org/wiki/Tjalling_Koopmanshttps://en.wikipedia.org/wiki/Nobel_prize_in_economicshttps://en.wikipedia.org/wiki/Linear_programming#cite_note-Sierksma2001-1https://en.wikipedia.org/wiki/Frank_Lauren_Hitchcockhttps://en.wikipedia.org/wiki/Frank_Lauren_Hitchcockhttps://en.wikipedia.org/wiki/Linear_programming#cite_note-Schrijver1998-2https://en.wikipedia.org/wiki/George_Dantzighttps://en.wikipedia.org/wiki/Simplex_algorithmhttps://en.wikipedia.org/wiki/John_von_Neumannhttps://en.wikipedia.org/wiki/John_von_Neumannhttps://en.wikipedia.org/wiki/Linear_programming#Dualityhttps://en.wikipedia.org/wiki/Game_theoryhttps://en.wikipedia.org/wiki/Linear_programming#cite_note-3


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ant,igs #rigina" e?am"e !as t# $ind the .est assignment #$ 70 e#"e t# 70 K#.s* %he

c#muting #!er re)uired t# test a"" the ermutati#ns t# se"ect the .est assignment is

'ast; the num.er #$ #ssi."e c#n$igurati#ns e?ceeds the num.er #$ artic"es in the

#.ser'a."e uni'erse* /#!e'er& it ta-es #n"y a m#ment t# $ind the #timum s#"uti#n .y

#sing the r#."em as a "inear r#gram and a"ying the sim"e? a"g#rithm* %he the#ry.ehind "inear r#gramming drastica""y reduces the num.er #$ #ssi."e s#"uti#ns that

must .e chec-ed*

%he "inear4r#gramming r#."em !as $irst sh#!n t# .e s#"'a."e in #"yn#mia" time

.y e#nid Ohachiyanin 1@7@& .ut a "arger the#retica" and ractica" .rea-thr#ugh in the

$ie"d came in 1@HG !hen9arendraOarmar-ar intr#duced a ne! interi#r4#int meth#d $#r

s#"'ing "inear4r#gramming r#."ems*

Uses

inear r#gramming is a c#nsidera."e $ie"d #$ #timi,ati#n $#r se'era" reas#ns* any

ractica" r#."ems in#erati#ns research can .e e?ressed as "inear r#gramming

r#."ems* ertain secia" cases #$ "inear r#gramming& such as net)or+ flo)r#."ems

and multicommodit, flo)r#."ems are c#nsidered im#rtant en#ugh t# ha'e generated

much research #n secia"i,ed a"g#rithms $#r their s#"uti#n* A num.er #$ a"g#rithms $#r

#ther tyes #$ #timi,ati#n r#."ems !#r- .y s#"'ing P r#."ems as su.4r#."ems*

/ist#rica""y& ideas $r#m "inear r#gramming ha'e insired many #$ the centra" c#ncets #$

#timi,ati#n the#ry& such as dualit,%decom"osition%and the im#rtance #$ convexit,and

its genera"i,ati#ns* i-e!ise& "inear r#gramming is hea'i"y used inmicr#ec#n#micsandc#many management& such as "anning& r#ducti#n& trans#rtati#n& techn#"#gy and

#ther issues* A"th#ugh the m#dern management issues are e'er4changing& m#st

c#manies !#u"d "i-e t# ma?imi,e r#$its #r minimi,e c#sts !ith "imited res#urces*

%here$#re& many issues can .e characteri,ed as "inear r#gramming r#."ems

+n rea" "i$e& "inear r#gramming is art #$ a 'ery im#rtant area #$ mathematics ca""ed#timi,ati#n techni)ues* %his $ie"d #$ study (#r at "east the a"ied resu"ts #$ it are usede'ery day in the #rgani,ati#n and a""#cati#n #$ res#urces* %hese rea" "i$e systems canha'e d#,ens #r hundreds #$ 'aria."es& #r m#re* +n a"ge.ra& th#ugh& y#u"" #n"y !#r- !ith

the sim"e (and graha."e t!#4'aria."e "inear case*

%he genera" r#cess $#r s#"'ing "inear4r#gramming e?ercises is t# grah the ine)ua"ities(ca""ed the c#nstraints t# $#rm a !a""ed4#$$ area #n thex%,4"ane (ca""ed the $easi.i"ityregi#n* %hen y#u $igure #ut the c##rdinates #$ the c#rners #$ this $easi.i"ity regi#n (thatis& y#u $ind the intersecti#n #ints #$ the 'ari#us airs #$ "ines& and test these c#rner

points in the formula (called the 9optimi'ation e1uation9$ for which youOre trying to

find the highest or lowest value.
https://en.wikipedia.org/wiki/Simplex_algorithmhttps://en.wikipedia.org/wiki/Leonid_Khachiyanhttps://en.wikipedia.org/wiki/Narendra_Karmarkarhttps://en.wikipedia.org/wiki/Interior-point_methodhttps://en.wikipedia.org/wiki/Operations_researchhttps://en.wikipedia.org/wiki/Microeconomicshttps://en.wikipedia.org/wiki/Simplex_algorithmhttps://en.wikipedia.org/wiki/Leonid_Khachiyanhttps://en.wikipedia.org/wiki/Narendra_Karmarkarhttps://en.wikipedia.org/wiki/Interior-point_methodhttps://en.wikipedia.org/wiki/Operations_researchhttps://en.wikipedia.org/wiki/Microeconomics


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UTHER EPLORATION

aLin!a+ P+4+a**in BLP

Method to achieve the best outcome (such as ma#imum profit or lower cost$ in a

mathematical model whose re1uirements are represented by linear relationships.It is a special case of mathematical programming.

A techni1ue for the optimi'ation of a linear objective function, subject to linear

e1uality and linear ine1uality constraints. @an be applied to various field of study. It is used in business and economies, but

can also be utili'ed for some engineering problems. Industries that us linear

programming models include transportation, energy, telecommunication, and

manufacturing. It has proved useful in diverse type of problems in planning,

routing, scheduling, assignment and design. Many practical problems in operations research can be e#presses as linear

programming problems. @ertain special cases of linear programming, such as

network flow problems and multicommodity flow problems are considered

important enough to have generated much research on speciali'ed algorithms for

their solution. Listorically, ideas from linear programming have inspired many of

the central concept of optimi'ation theory, such as duality, decomposition and the

importance of conve#ity and its generali'ations.

E@a*6!":

i% A manufacturer produces two products, # and y, with two machines, Aand K.he cost of producting he cost of producting

each unit of # is7 each unit of y is

For machine A 7= minutes For machine A 7M;-

For machine K 73 minutes For machine K 7M33

ii% A manufacturer produces two products, # and y, with two machines, Aand K.he cost of producting he cost of producting

each unit of # is7 each unit of y is

For machine A 7= minutes For machine A 7M;-

For machine K 73 minutes For machine K 7M33


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@abinet type 0rice (M$/unit Area(m;$ olme(m3$

# 5 .B .

y ; . 5.;

i. (a$ #7y 8 ;73 5# ;y 6 5- .B# .y 6 :.;

x

y 82

3 # ;y 6 5- B# y 6 :;

;y 6 3# 3# -y 6 3B

ii. Method 5

olume ) .# 5.;yH1uation 7 .# 5.;y ) 5.E;


30/37

0oint @abinet # @abinet y otal volume).# 5.;y

(-, =$ - = E.;

(=, -$ = - .

(B, -$ B - E.B

(:, 3$ : 3 E.;

(, 3$ 3(E, ;$ E ; E.B

(5, 5$ 5 5 E.;

From the table above, the ma#imum storage volume is .($ 5.;(3$

) 5m3

iii.

0oint @abinet # @abinet y otal cost(M$)5# ;y

(-, 5$ - 5 B

(-, ;$ - ;

(-, 3$ - 3 5(-, -$ - - 5;

(-, =$ - = 5-

(=, 5$ = 5 :

(=, ;$ = ; E

(=, 3$ = 3 55

(=, -$ = - 53

(B, 5$ B 5

(B, ;$ B ; 5

(B, 3$ B 3 5;

(B, -$ B - 5-

(:, 5$ : 5 E

(:, ;$ : ; 55

(:, 3$ : 3 53

(, 5$ 5 5

(, ;$ ; 5;

(, 3$ 3 5-

(E, 5$ E 5 55

(E, ;$ E ; 53

he table above shows all the possible combination of the cabinets that Aaron plans to

buy and the cost of each combination.

iv. If I were Aaron, I would choose cabinet (, 3$ because the cost is e#actly M 5-

and it is also ma#imi'e the area :.;m;and volume 5m3.

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PART 1


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PART 2


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PART /


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URTHEREPLORATION

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ADDITIONAL MATH(1).docx