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6.1 Optimization
121
tominimizeis
f(x)=2x+2
100
x
sincetheperimeteristwicethelengthplustwicethewidthoftherectangle.Notallvalues
ofxmakesenseinthisproblem:lengthsofsidesofrectanglesmustbepositive,sox>0.
Ifx>0thensois100=x,soweneednosecondconditiononx.
Wenextndf
0
(x)andsetitequaltozero:0=f
0
(x)=2 200=x
2
. Solvingf
0
(x)=0
forx givesusx=10. Weareinterestedonlyinx>0,so o onlythe value x=10 isof
interest.Sincef
0
(x)isdenedeverywhereontheinterval(0;1),therearenomorecritical
values, andthere e are noendpoints. Isthere e alocalmaximum, , minimum,orneitherat
x =10? The e second derivative isf
00
(x)= 400=x
3
,and f
00
(10) >0, , so o there isa local
minimum. Since e thereisonlyonecriticalvalue,thisisalsotheglobalminimum,sothe
rectanglewithsmallestperimeteristhe1010square.
EXAMPLE6.1.8
Youwanttosellacertainnumbernofitemsinordertomaximize
yourprot. Marketresearchtellsyouthatifyousetthepriceat\$1.50,youwillbeable
tosell5000items,andforevery10centsyoulowerthepricebelow\$1.50youwillbeable
tosellanother1000items. Supposethatyourxedcosts(\start-upcosts")total\$2000,
andtheperitemcostofproduction(\marginalcost")is\$0.50. Findthepricetosetper
itemandthe numberofitemssoldinorderto maximize prot,andalsodetermine the
maximumprotyoucanget.
Therststepistoconverttheproblemintoafunctionmaximizationproblem. Since
we wanttomaximizeprotbysetting theprice peritem,weshouldlookforafunction
P(x)representingtheprotwhenthepriceperitemisx.Protisrevenueminuscosts,and
revenueisnumberofitemssoldtimesthepriceperitem,sowegetP =nx 2000 0:50n.
Thenumberofitemssoldisitselfafunctionofx,n=5000+1000(1:5 x)=0:10,because
(1:5 x)=0:10isthenumberofmultiplesof10centsthatthepriceisbelow\$1.50. . Now
wesubstitutefornintheprotfunction:
P(x)=(5000+1000(1:5 x)=0:10)x 2000 0:5(5000+1000(1:5 x)=0:10)
= 10000x
2
+25000x 12000
Wewanttoknowthemaximumvalueofthisfunctionwhenxisbetween0and1:5. The
derivative isP
0
(x) = 20000x+25000,which h is s zero o whenx x = 1:25. Since e P
00
(x) =
20000<0,theremustbealocalmaximumatx=1:25,andsincethisistheonlycritical
value it must be a a global l maximum as well. (Alternately, , we could d compute P(0) ) =
12000, P(1:25)=3625,andP(1:5) ) =3000 andnote thatP(1:25) isthe maximumof
these.) Thusthemaximumprot t is\$3625,attainedwhenwesetthepriceat\$1.25 and
sell7500items.
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122
Chapter 6 6 Applicationsofthe e Derivative
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y=a
(x;x
2
)
............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............
Figure6.1.3
Rectangleinaparabola.
EXAMPLE6.1.9
Findthelargestrectangle(thatis,therectanglewithlargestarea)
thattsinsidethegraphoftheparabolay=x
2
belowtheliney=a(aisanunspecied
constant value), , with the e top side of the rectangle on the horizontal line y y = = a; see
gure6.1.3.)
WewanttondthemaximumvalueofsomefunctionA(x)representingarea.Perhaps
the hardest part of thisproblemisdeciding what x should represent. The e lower r right
cornerofthe rectangleisat (x;x
2
), andonce e thisischosentherectangle iscompletely
determined. Sowecanletthe e xinA(x) bethex oftheparabolaf(x)=x
2
. Thenthe
areaisA(x)=(2x)(a x
2
)= 2x
3
+2ax. WewantthemaximumvalueofA(x)whenxis
in[0;
p
a].(Youmightobjecttoallowingx=0orx=
p
a,sincethenthe\rectangle"has
eithernowidthornoheight,soisnot\really"arectangle. Buttheproblemissomewhat
easierifwesimplyallowsuchrectangles,whichhavezeroarea.)
Setting0=A
0
(x)= 6x
2
+2awegetx=
p
a=3astheonlycriticalvalue. Testing
thisand thetwo endpoints, wehaveA(0)=A(
p
a) =0 andA(
p
a=3) =(4=9)
p
3a
3=2
.
Themaximumareathusoccurswhentherectanglehasdimensions2
p
a=3(2=3)a.
EXAMPLE 6.1.10
If yout thelargest possible cone insideasphere,whatfraction
ofthe volume of thesphereisoccupiedbythe cone? (Here e by\cone"we meana right
circularcone,i.e.,aconeforwhichthebaseisperpendiculartotheaxisofsymmetry,and
forwhichthe cross-section cutperpendiculartothe axisofsymmetryatanypoint isa
circle.)
theconeinsidethesphere.Whatwewanttomaximizeisthevolumeofthecone:r
2
h=3.
HereRisaxedvalue,butrandhcanvary. Namely,wecouldchoosertobeaslargeas
possible|equaltoR|bytakingtheheightequaltoR;orwecouldmakethecone’sheight
hlargerattheexpenseofmakingralittlelessthanR. Seethecross-sectiondepictedin
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6.1 Optimization
123
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(h R;r)
Figure 6.1.4
Coneinasphere.
gure6.1.4.Wehavesituatedthepictureinaconvenientwayrelativetothexandyaxes,
namely,withthecenterofthesphereattheoriginandthevertexoftheconeatthefar
leftonthex-axis.
Noticethatthefunctionwewanttomaximize,r
2
h=3,dependsontwovariables.This
isfrequentlythecase,butoftenthetwovariablesarerelatedinsomewaysothat\really"
thereisonlyonevariable. Soournextstepistondtherelationshipanduseittosolve
foroneofthevariablesintermsoftheother,soastohaveafunctionofonlyonevariable
tomaximize. Inthisproblem,theconditionisapparentinthegure:theuppercornerof
thetriangle,whosecoordinatesare(h R;r),mustbeonthecircleofradiusR. . Thatis,
(h R)
2
+r
2
=R
2
:
We cansolve forhintermsofr orforr intermsofh. Eitherinvolvestakingasquare
root,butwenoticethatthevolumefunctioncontainsr
2
,notrbyitself,soitiseasiestto
solveforr
2
directly:r
2
=R
2
(h R)
2
. Thenwesubstitutetheresultintor
2
h=3:
V(h)=(R
2
(h R)
2
)h=3
3
h
3
+
2
3
h
2
R
We want to maximize V(h) ) when n h is s between n 0 and 2R. Nowwe e solve 0 = = f
0
(h) =
h
2
+(4=3)hR,getting h h =0 0 orh h =4R=3. We e compute V(0) ) =V(2R) ) = = 0 0 and
V(4R=3) =(32=81)R
3
. The e maximumisthelatter; ; sincethevolumeof thesphereis
(4=3)R
3
,thefractionofthesphereoccupiedbytheconeis
(32=81)R3
(4=3)R3
=
8
27
30%:
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124
Chapter 6 6 Applicationsofthe e Derivative
EXAMPLE6.1.11
Youaremakingcylindricalcontainerstocontainagivenvolume.
Suppose that thetopandbottomare made ofa materialthat isN N timesasexpensive
(costperunitarea)asthematerialusedforthelateralsideofthecylinder. Find(interms
thecontainers.
Letusrstchooseletterstorepresentvariousthings: hfortheheight,rforthebase
ofthecylinder;V andcareconstants,handr r arevariables. . Nowwecanwritethecost
ofmaterials:
c(2rh)+Nc(2r
2
):
Again we have two variables; the relationship is provided by y the xed d volume of the
cylinder:V =r
2
h.Weusethisrelationshiptoeliminateh(wecouldeliminater,butit’s
alittleeasierifweeliminateh,whichappearsinonlyoneplaceintheaboveformulafor
cost).Theresultis
f(r)=2cr
V
r2
+2Ncr
2
=
2cV
r
+2Ncr
2
:
Wewanttoknowtheminimumvalue ofthisfunctionwhenr isin(0;1). We e nowset
0=f
0
(r)= 2cV=r
2
+4Ncr,giving r =
3
p
V=(2N). Sincef
00
(r)=4cV=r
3
+4Nc
ispositivewhenrispositive,thereisalocalminimumatthecriticalvalue,andhencea
globalminimumsincethereisonlyonecriticalvalue.
Finally,sinceh=V=(r
2
),
h
r
=
V
r3
=
V
(V=(2N))
=2N;
sothe minimumcost occurswhen theheighth is2N timestheradius. If,forexample,
isequaltothediameter).
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x
a x
A
D
B
C
b
Figure 6.1.5
Minimizingtraveltime.
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6.1 Optimization
125
EXAMPLE 6.1.12
Suppose youwant toreach a point Athat islocatedacrossthe
letw,which islessthanv,beyourspeedonthe sand. Rightnowyouareat t thepoint
acrossthesandinordertominimizeyourtraveltimetoA?
LetxbethedistanceshortofCwhereyouturno,i.e.,thedistancefromBtoC.We
wantto minimize thetotaltraveltime. Recallthatwhentravelingatconstant t velocity,
timeisdistancedividedbyvelocity.
Youtravelthedistance
DBatspeedv,andthenthedistance
BAatspeedw. Since
DB =a x x and,bythePythagorean theorem,
BA=
p
x2+b2,thetotaltime forthe
tripis
f(x)=
a x
v
+
p
x2+b2
w
:
We want to ndthe minimumvalue off f whenx x isbetween0 anda. Asusualwe e set
f
0
(x)=0andsolveforx:
0=f
0
(x)=
1
v
+
x
w
p
x2+b2
w
p
x2+b=vx
w
2
(x
2
+b
2
)=v
2
x
2
w
2
b
2
=(v
2
w
2
)x
2
x=
wb
p
v2 w2
Notice that t adoesnot t appear r in n the e last expression, but aisnot t irrelevant, , since e we
areinterestedonlyincriticalvaluesthatare in[0;a],andwb=
p
v2 wiseitherinthis
intervalornot. Ifitis,wecanusethesecondderivativetotestit:
f
00
(x)=
b
2
(x2+b2)3=2w
:
Sincethisisalwayspositivethereisalocalminimumatthecriticalpoint,andsoitisa
globalminimumaswell.
Ifthecriticalvalueisnotin[0;a]itislargerthana. Inthiscasetheminimummust
occuratoneoftheendpoints.Wecancompute
f(0)=
a
v
+
b
w
f(a)=
p
a2+b2
w
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126
Chapter6 ApplicationsoftheDerivative
butit isdicult to determine which of these issmallerbydirect comparison. If,asis
likelyinpractice,weknowthevaluesofv,w,a,andb,thenitiseasytodeterminethis.
Withalittlecleverness,however,wecandeterminetheminimumingeneral.Wehaveseen
thatf
00
(x)isalwayspositive,sothederivativef
0
(x)isalwaysincreasing. Weknowthat
atwb=
p
v2 wthederivativeiszero,soforvaluesofxlessthanthatcriticalvalue,the
derivativeisnegative.Thismeansthatf(0)>f(a),sotheminimumoccurswhenx=a.
Sotheupshotisthis: IfyoustartfartherawayfromCthanwb=
p
v2 wthenyou
p
v2 wfrompointC.
IfyoustartcloserthanthistoC,youshouldcutdirectlyacrossthesand.
Summary|Stepstosolveanoptimizationproblem.
1. Decide e what the e variables are and d what t the e constants s are, , draw w a diagram if
appropriate,understandclearlywhatitisthatistobemaximizedorminimized.
2. Write a a formula a forthe function n for r which h you u wish h to o nd the e maximum m or
minimum.
3. Expressthatformulaintermsofonlyonevariable,thatis,intheformf(x).
4. Setf
0
(x)=0andsolve. Checkallcriticalvaluesandendpointstodeterminethe
extremevalue.
Exercises 6.1.
1. Letf(x)=
1+4x x
2
forx3
(x+5)=2
forx>3
Findthemaximumvalueandminimumvalues off(x)forxin[0;4]. . Graphf(x)tocheck
2. Findthedimensionsoftherectangleoflargestareahavingxedperimeter100. )
3. FindthedimensionsoftherectangleoflargestareahavingxedperimeterP.)
4. Aboxwithsquarebaseandnotopistoholdavolume100.Findthedimensionsofthebox
thatrequirestheleastmaterialforthevesides. Alsondtheratioofheighttosideofthe
base.)
5. Aboxwithsquarebaseistoholdavolume200. . Thebottomandtopareformedbyfolding
in aps fromallfour sides,sothat thebottomandtopconsist oftwolayersofcardboard.
Findthedimensionsoftheboxthatrequirestheleastmaterial. Alsondtheratioofheight
tosideofthebase. )
6. AboxwithsquarebaseandnotopistoholdavolumeV.Find(intermsofV)thedimensions
oftheboxthatrequirestheleastmaterialforthevesides. Alsondtheratioofheightto
sideofthebase. (ThisratiowillnotinvolveV.) )
7. Youhave100feetoffencetomakearectangularplayareaalongsidethewallofyourhouse.
Thewallofthehouseboundsoneside. Whatisthelargestsizepossible(insquarefeet)for
theplayarea? )
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6.1 Optimization
127
8. Youhavel l feet offencetomake arectangular playareaalongsidethewallofyour house.
Thewallofthehouseboundsoneside. Whatisthelargestsizepossible(insquarefeet)for
theplayarea? )
9. Marketingtellsyouthatifyousetthepriceofanitemat\$10thenyouwillbeunabletosell
it,butthatyoucansell500itemsforeachdollarbelow\$10thatyousettheprice.Suppose
yourxedcoststotal\$3000,andyourmarginalcostis\$2peritem.Whatisthemostprot
youcanmake?)
10. Findtheareaofthe e largestrectanglethat tsinsideasemicircleof radius 10(onesideof
therectangleisalongthediameterofthesemicircle). )
rectangleisalongthediameterofthesemicircle). )
12. Foracylinderwithsurfacearea50,includingthetopandthebottom,ndtheratioofheight
13. ForacylinderwithgivensurfaceareaS,includingthetopandthebottom,ndtheratioof
14. Youwanttomakecylindricalcontainers s tohold1liter(1000cubiccentimeters)usingthe
andthis canbedonewithnomaterialwasted. . However,thetopandbottomarecutfrom
squaresofside 2r, so that 2(2r)
2
=8r
2
of materialis needed(rather than 2r
2
,whichis
thetotalareaofthetopandbottom). Findthedimensionsofthecontainerusingtheleast
15. You u want t to make cylindrical l containers of f a a given volume V using g the least t amount t of
constructionmaterial. Thesideismadefromarectangular r pieceofmaterial,andthiscan
bedonewithnomaterialwasted. However,thetopandbottomarecutfromsquaresofside
2r,sothat2(2r)
2
=8r
2
ofmaterialisneeded(ratherthan2r
2
,whichisthetotalareaof
16. Givenarightcircularcone,youputanupside-downconeinsideitsothatitsvertexisatthe
centerofthebaseofthelargerconeanditsbaseisparalleltothebaseofthelargercone.If
youchoosetheupside-downconetohavethelargest possiblevolume,whatfractionofthe
volume ofthelarger conedoesit occupy? ? (Let t H H andRbetheheightandbaseradiusof
similartrianglestogetanequationrelatinghandr.) )
17. Inexample6.1.12,whathappensifwv(i.e.,yourspeedonsandisatleastyourspeedon
18. Acontainerholding g axedvolumeis beingmade intheshapeofacylinderwithahemi-
sphericaltop. (Thehemisphericaltophasthesameradiusasthecylinder.) ) Findtheratioof
unitareaofthetopistwiceasgreatasthecostperunitareaoftheside,andthecontainer
is madewithnobottom; ; (b)thesameas s in(a),except thatthe container ismadewitha
circularbottom,forwhichthecost perunitareais1.5times thecostper unitareaofthe
side.)
19. Apieceofcardboardis1meterby1=2meter. Asquareis s tobecutfromeachcornerand
the sides foldedup to make an open-top box. What t are the dimensions of the box with
maximumpossiblevolume? )
128
Chapter6 ApplicationsoftheDerivative
20. (a)Asquarepieceof f cardboardofside ais usedtomakeanopen-topbox by cuttingout
asmallsquarefromeachcornerandbendingup the sides. How w large asquare should be
cutfromeachcornerinorderthattheboxhavemaximumvolume?(b)Whatifthepieceof
cardboardusedtomaketheboxisarectangleofsidesaandb?)
21. Awindow w consistsofarectangularpieceofclearglasswithasemicircularpieceofcolored
glassontop;thecoloredglasstransmitsonly1=2asmuchlightperunitareaasthetheclear
glass. Ifthe e distancefromtoptobottom(across boththe rectangle andthesemicircle) is
2metersandthewindowmaybenomorethan1.5meterswide,ndthedimensionsofthe
rectangularportionofthewindowthatletsthroughthemostlight. )
22. Awindow w consistsofarectangularpieceofclearglasswithasemicircularpieceofcolored
glass ontop. . Supposethatthecoloredglasstransmitsonlyktimesasmuchlightper r unit
area as the clear glass(k is between0 and1). Ifthedistance e fromtoptobottom(across
boththerectangleandthesemicircle)isaxeddistanceH,nd(intermsofk)theratioof
verticalsidetohorizontalsideoftherectangleforwhichthewindowletsthroughthemost
light.)
23. You u are designing a poster to contain a xed amount Aof printing (measuredinsquare
centimeters)andhavemarginsofacentimetersatthetopandbottomandbcentimetersat
thesides. Findtheratioofverticaldimensiontohorizontaldimensionoftheprintedareaon
theposterifyouwanttominimizetheamountofposterboardneeded. )
24. Thestrengthofarectangularbeamisproportionaltotheproductofitswidthwtimesthe
squareofits depth h d. Findthe e dimensions ofthe strongest beam that canbe cutfroma
.
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j
j
d
j
j
#
w  !
Figure6.1.6
Cuttingabeam.
25. What t fractionofthevolumeofasphereistakenupbythelargest cylinderthat canbet
insidethesphere? )
26. TheU.S.postocewillacceptaboxforshipmentonlyifthesumofthelengthandgirth
(distancearound)isatmost108in.Findthedimensionsofthelargestacceptableboxwith
squarefrontandback. )
27. Findthedimensions s of thelightestcylindricalcancontaining 0.25liter (=250cm
3
) ifthe
usedfortheside. )
28. Aconicalpapercupistohold1=4ofaliter. . Findtheheightandradius s oftheconewhich
minimizesthe amountof paper neededtomakethecup. . Use e the formular
p
r+hfor
theareaofthesideofacone.)
oftheconewhichminimizestheamountofpaperneededtomakethecup. Usetheformula
r
p
r2+hfortheareaofthesideofacone,calledthelateral areaofthecone. )
6.2 RelatedRates
129
30. Ifyouttheconewiththelargestpossiblesurfacearea(lateralareaplusareaofbase)into
asphere,whatpercentofthevolumeofthesphereisoccupiedbythecone? )
31. Two o electrical charges, one a positive charge A of magnitude e a a and d the other r a negative
chargeBofmagnitudeb,arelocatedadistancecapart. ApositivelychargedparticleP P is
situatedonthelinebetweenAandB.FindwhereP shouldbe e put sothat thepullaway
fromAtowardsBisminimal. Hereassumethattheforcefromeachchargeisproportional
tothestrengthofthesourceandinversely proportionaltothesquareofthedistancefrom
thesource.)
32. Findthefractionoftheareaofatrianglethatisoccupiedbythelargestrectanglethatcan
bedrawninthetriangle(withoneofitssides alongasideofthetriangle). . Showthatthis
fractiondoesnotdependonthedimensionsofthegiventriangle. )
33. How w are your answers to Problem 9aected if thecost per itemfor the xitems,instead
ofbeingsimply\$2,decreasesbelow\$2inproportiontox(becauseofeconomyofscaleand
volumediscounts)by1centforeach25itemsproduced?)
introuble. Youcanrunataspeedv
1
onlandandataslowerspeedv
2
inthewater. Your
perpendiculardistancefromthesideofthepoolisa,thechild’sperpendiculardistanceisb,
andthedistancealongthesideofthepoolbetweentheclosestpointtoyouandtheclosest
point to the childis s c c (see the gure below). Without t stopping to do any calculus, , you
instinctivelychoosethequickestroute(showninthegure)andsavethechild. Ourpurpose
istoderivearelationbetweentheangle
1
yourpathmakeswiththeperpendiculartotheside
ofthepoolwhenyou’reonland,andtheangle
2
yourpathmakeswiththeperpendicular
whenyou’reinthewater. Todothis,letxbethedistancebetweentheclosestpointtoyou
at thesideofthepoolandthe pointwhereyouenter thewater. . Writethetotaltimeyou
run(onlandandinthewater)intermsofx(andalsotheconstantsa;b;c;v
1
;v
2
). Thenset
thederivativeequaltozero. Theresult,called\Snell’slaw"orthe\lawofrefraction,"also
governsthebendingoflightwhenitgoesintowater.)
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x
c x
a
b
Figure 6.1.7