﻿
6.1 Optimization
117
areaas theclearglass(k is between0and1). . Ifthe e distancefromtopto bottom(across
boththerectangleandthesemicircle)isaﬁxeddistanceH,ﬁnd(intermsofk)theratioof
verticalsidetohorizontalsideoftherectangleforwhichthewindowletsthroughthemost
light.
23. You u are designing a poster r to contain n a ﬁxedamount A of printing (measured in square
centimeters)andhavemarginsofacentimetersatthetopandbottomandbcentimetersat
thesides. Findtheratioofverticaldimensiontohorizontaldimensionoftheprintedareaon
theposterifyouwanttominimizetheamountofposterboardneeded.
24. Thestrengthofarectangularbeamisproportionaltotheproductofitswidthwtimesthe
square ofits depthd. . Findthe e dimensionsofthe strongest beamthat canbe cutfroma
.
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←−
w
−→
Figure 6.6
Cuttingabeam.
25. Whatfractionofthevolumeofasphereistakenupby y thelargest cylinderthat canbeﬁt
insidethesphere?
26. TheU.S.post t oﬃcewillacceptaboxforshipmentonlyifthesumofthelengthandgirth
(distancearound)isatmost108in.Findthedimensionsofthelargestacceptableboxwith
squarefrontandback.
27. Findthe e dimensions of thelightestcylindricalcancontaining 0.25liter (=250cm
3
) if the
usedfortheside.
minimizes theamount of paper neededtomakethe cup. . Use e theformulaπr
r+hfor
theareaofthesideofacone.
oftheconewhichminimizestheamountofpaperneededtomakethecup. Usetheformula
πr
r+hfortheareaofthesideofacone,calledthelateral areaofthecone.
30. Ifyouﬁttheconewiththelargestpossiblesurfacearea(lateralareaplusareaofbase)into
asphere,whatpercentofthevolumeofthesphereisoccupiedbythecone?
31. Two o electrical charges, one a positive charge Aof magnitude e a a and d the other r a negative
situatedontheline betweenAandB.Findwhere P P shouldbe e put so thatthe pullaway
fromAtowardsBisminimal. Hereassumethattheforcefromeachchargeisproportional
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118
Chapter6 Applications s oftheDerivative
tothestrengthof the sourceandinverselyproportionaltothesquareofthedistancefrom
thesource.
32. Findthefractionoftheareaofatrianglethatisoccupiedbythelargestrectanglethatcan
bedrawninthetriangle(withoneofitssidesalongasideofthetriangle). Showthatthis
fractiondoesnotdependonthedimensionsofthegiventriangle.
33. How w areyour answers toProblem aﬀectedif the costper itemfor the x items, , instead
ofbeingsimply\$2,decreasesbelow\$2inproportiontox(becauseofeconomyofscaleand
volumediscounts)by1centforeach25itemsproduced?
introuble. Youcanrunataspeedv
1
onlandandataslowerspeedv
2
inthewater. Your
perpendiculardistancefromthesideofthepoolisa,thechild’sperpendiculardistanceisb,
andthedistancealongthesideofthepoolbetweentheclosestpointtoyouandtheclosest
point tothe e child d is s c (see the ﬁgure below). . Without t stopping g to do any calculus, you
instinctivelychoosethequickestroute(shownintheﬁgure)andsavethechild. Ourpurpose
istoderivearelationbetweentheangleθ
1
yourpathmakeswiththeperpendiculartotheside
ofthepoolwhenyou’reonland,andtheangleθ
2
yourpathmakeswiththeperpendicular
whenyou’reinthewater. Todothis,letxbethedistancebetweentheclosestpointtoyou
atthesideofthepoolandthepointwhere youenterthewater. . 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
Figure6.7
Suppose we have two variables s x x and y (in most problemsthe letters s willbediﬀerent,
but for r nowlet’s s use x x and y) ) which h are e both h changing with time. . A A “related rates”
problemisaprobleminwhichweknowoneoftheratesofchangeatagiveninstant—say,
x˙ =dx/dt—andwe e want toﬁndthe other rate e ˙y=dy/dtatthatinstant. . (Theuse e of
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6.2 RelatedRates
119
x˙ to o meandx/dt goesbacktoNewton andisstillused for thispurpose, , especiallyby
physicists.)
Ifyiswrittenintermsofx,i.e.,y=f(x),thenthisiseasytodousingthechainrule:
y˙=
dy
dt
=
dy
dx
·
dx
dt
=
dy
dx
x˙.
Thatis,ﬁndthederivativeoff(x),pluginthevalueofxattheinstantinquestion,and
multiplybythegivenvalueof ˙x=dx/dttoget ˙y=dy/dt.
EXAMPLE6.13
Supposeanobjectismovingalongapathdescribedbyy=x
2
,that
is,itismovingonaparabolicpath. Ataparticulartime,sayt=5,thexcoordinateis
6andwemeasurethespeedatwhichthexcoordinateoftheobjectischangingandﬁnd
thatdx/dt=3. Atthesametime,howfastistheycoordinatechanging?
Usingthechainrule,dy/dt=2x·dx/dt.Att=5weknowthatx=6anddx/dt=3,
sody/dt=2·6·3=36.
Inmanycases,particularlyinterestingones,xandywillberelatedinsomeotherway,
for examplex=f(y),or F(x,y) =k,or perhapsF(x,y)=G(x,y), , where e F(x,y)and
G(x,y)areexpressionsinvolvingbothvariables. In n allcases, youcan solve the related
ratesproblemby taking g the derivative of both sides, , plugging g in all l the e known n values
(namely,x,y,andx˙),andthensolvingfory˙.
Tosummarize,herearethestepsindoingarelatedratesproblem:
1. Decidewhatthetwovariablesare.
2. Findanequationrelatingthem.
3. Taked/dtofbothsides.
4. Pluginallknownvaluesattheinstantinquestion.
5. Solvefortheunknownrate.
EXAMPLE 6.14
Aplaneisﬂyingdirectlyawayfromyouat500mphatanaltitude
of3miles. Howfastistheplane’sdistancefromyouincreasingatthemomentwhenthe
planeisﬂyingoverapointontheground4milesfromyou?
Toseewhat’sgoingon,weﬁrstdrawaschematicrepresentationofthesituation.
Becausetheplaneisinlevelﬂightdirectlyawayfromyou,therateatwhichxchanges
isthespeedoftheplane,dx/dt=500. Thedistancebetweenyouandtheplaneisy;it
isdy/dtthatwewishtoknow. BythePythagoreanTheoremweknowthatx
2
+9=y
2
.
Takingthederivative:
2xx˙=2yy˙.
120
Chapter6 Applications s oftheDerivative
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Figure 6.8
Recedingairplane.
Weareinterestedinthetimeatwhichx=4;atthistimeweknowthat4
2
+9=y
2
,so
y=5. Puttingtogetheralltheinformationwehaveweget
2(4)(500)=2(5)˙y.
Thus, ˙y=400mph.
EXAMPLE 6.15
Youareinﬂatingasphericalballoonattherateof7cm3/sec. How
dr/dt. ThetwovariablesarerelatedbymeansoftheequationV V =4πr3/3. Takingthe
derivative ofbothsidesgivesdV/dt=4πr
2
˙r. Wenowsubstitutethe e valuesweknowat
theinstantinquestion: 7=4π4
2
r˙,sor˙=7π/64cm/sec.
EXAMPLE 6.16
Water ispouredintoaconicalcontainerattherateof10cm3/sec.
Howfastisthewaterlevelrisingwhenthewateris4cmdeep(atitsdeepestpoint)?
volumeareallincreasingaswaterispouredintothecontainer.Thismeansthatweactually
havethreethingsvaryingwithtime: thewaterlevelh(theheightoftheconeofwater),
thevolumeofwaterV. ThevolumeofaconeisgivenbyV =πr
2
h/3. WeknowdV/dt,
andwe want dh/dt. . At t ﬁrst something seemsto be wrong: : we e have athird variable r
whoseratewedon’tknow.
Butthedimensionsoftheconeofwatermusthavethesameproportionsasthoseof
thecontainer. Thatis,becauseofsimilartriangles,r/h=10/30sor=h/3. Nowwecan
eliminaterfromtheproblementirely:V =π(h/3)
2
h/3=πh
3
/27.Wetakethederivative
ofbothsidesandpluginh=4anddV/dt=10,obtaining10=(3π·4
2
/27)(dh/dt).Thus,
dh/dt=90/(16π)cm/sec.
6.2 RelatedRates
121
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←−
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−→
↑|
30
|
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|↓
r
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Figure6.9
Conicalwatertank.
EXAMPLE6.17
Aswingconsistsofaboardattheendofa10ftlongrope.Thinkof
theboardasapointP attheendoftherope,andletQbethepointofattachmentatthe
otherend.SupposethattheswingisdirectlybelowQattimet=0,andisbeingpushed
bysomeonewhowalksat6ft/secfromlefttoright.Find(a)howfasttheswingisrising
after1sec;(b)theangularspeedoftheropeindeg/secafter1sec.
that the person pushing the swing ismoving horizontallyat a rate we know. . Inother
words, the e horizontal l coordinate e of P P isincreasing g at 6 ft/sec. . In n the e xy-plane let us
maketheconvenientchoiceofputtingtheoriginatthelocationofP attimet=0,i.e.,a
distance10directlybelowthepointofattachment. Thentherateweknowisdx/dt,and
inpart(a)theratewewantisdy/dt(therateatwhichP isrising). Inpart(b)therate
wewantis
˙
swungfromthe vertical. . (Actually,since e wewant ouranswer indeg/sec,atthe endwe
(a) Fromthediagramweseethatwehavearighttriangle whoselegsare xand10−y,
andwhosehypotenuseis10. Hencex
2
+(10−y)
2
=100. Takingthederivativeofboth
sidesweobtain:2xx˙+2(10−y)(0−y˙)=0.Wenowlookatwhatweknowafter1second,
namelyx = 6 6 (because x x started d at t 0 0 and hasbeen increasing at the e rate e of 6 ft/sec
for 1sec),y=2(becauseweget 10−y=8fromthePythagoreantheoremappliedto
thetriangle withhypotenuse10andleg6),andx˙ ˙ =6. . Puttinginthesevaluesgivesus
2·6·6−2·8y˙=0,fromwhichwecaneasilysolvefor ˙y: y˙=4.5ft/sec.
122
Chapter6 Applications s oftheDerivative
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x
y
θ
Figure 6.10
Swing.
(b) Here our r two variables are x x and θ, , so o we e want to o use e the e same right t triangle e as
inpart (a),butthistime relateθ θ tox. . Sincethe e hypotenuseisconstant(equalto10),
the best wayto do thisisto use the sine: : sinθ θ =x/10. . Taking g derivativeswe obtain
(cosθ)
˙
θ=0.1˙x.Attheinstantinquestion(t=1sec),whenwehavearighttrianglewith
sides6–8–10,cosθ=8/10andx˙=6. Thus(8/10)
˙
θ=6/10,i.e.,
˙
orapproximately43deg/sec.
Wehaveseenthatsometimesthereareapparentlymorethantwovariablesthatchange
withtime, but t inrealitythere are just two, , astheotherscanbe e expressed in termsof
justtwo. Butsometimestherereallyareseveralvariablesthatchangewithtime;aslong
asyouknowtheratesofchangeofallbutoneofthemyoucanﬁndtherateofchangeof
theremainingone. Asinthecasewhentherearejusttwovariables,takethe e derivative
ofboth sidesofthe equationrelating allofthe variables,and thensubstitute allofthe
knownvaluesandsolvefortheunknownrate.
EXAMPLE 6.18
80km/hr,whilecarBis15kilometerstotheeastofP andtravelingat100km/hr. How
fastisthedistancebetweenthetwocarschanging?
Leta(t)bethedistanceofcarAnorthofPattimet,andb(t)thedistanceofcarBeast
ofP attimet,andletc(t)thedistancefromcarAtocarBattimet.BythePythagorean
Theorem,c(t)
2
=a(t)
2
+b(t)
2
.Takingderivativesweget2c(t)c
(t)=2a(t)a
(t)+2b(t)b
(t),
6.2 RelatedRates
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Figure6.11
Carsmovingapart.
so
˙c=
a˙a+b
˙
b
c
=
a˙a+b
˙
b
a2+b2
.
Substitutingknownvaluesweget:
˙c=
10·80+15·100
102+152
=
460
13
≈127.6km/hr
atthetimeofinterest.
Noticehowthisproblemdiﬀersfromexample6.14.Inbothcaseswestartedwiththe
PythagoreanTheoremandtookderivativesonbothsides. However,inexample6.14one
ofthesideswasaconstant(thealtitudeoftheplane),andsothederivativeofthesquare
ofthatsideofthetrianglewassimplyzero. Inthisexample,ontheotherhand,allthree
sidesoftherighttrianglearevariables,eventhoughweareinterestedinaspeciﬁcvalue
ofeachsideofthetriangle(namely,whenthesideshave lengths10and15). . Makesure
thatyouunderstandatthestartoftheproblemwhatarethevariablesandwhatarethe
constants.
124
Chapter6 Applications s oftheDerivative
Exercises
1. Acylindricaltank k standingupright (withone circular base onthe ground) has radius 20
cm. Howfastdoes s thewaterlevelinthetankdropwhenthewateris beingdrainedat25
cm
3
/sec?
2. A A cylindrical tank standing g upright t (with h one circular base on the ground) ) has s radius 1
meter.Howfastdoesthewaterlevelinthetankdropwhenthewaterisbeingdrainedat3
literspersecond?
3. Aladder13meterslongrestsonhorizontalgroundandleans s againsta verticalwall. . The
4. Aladder13meterslongrestsonhorizontalgroundandleans s againsta verticalwall. . The
5. Arotatingbeaconislocated2milesoutinthewater. . LetAbethepointontheshorethat
isclosesttothebeacon. Asthebeaconrotatesat10rev/min,thebeamoflightsweepsdown
theshoreonceeachtimeitrevolves.Assumethattheshoreisstraight.Howfastisthepoint
wherethebeamhitstheshoremovingataninstantwhenthebeamislightingupapoint2
milesalongtheshorefromthepointA?
6. Abaseballdiamondisasquare90ftonaside. . Aplayerrunsfromﬁrstbasetosecondbase
at 15ft/sec. . Atwhatrate e isthe player’sdistance fromthirdbasedecreasingwhensheis
halfwayfromﬁrsttosecondbase?
7. Sandispouredontoasurfaceat15cm
3
/sec,formingaconicalpilewhosebasediameteris
alwaysequaltoitsaltitude. Howfastisthealtitudeofthepileincreasingwhenthepileis3
cmhigh?
theotherendpassing througharingattachedtothedockat apoint5 ft higherthanthe
frontoftheboat. Theropeisbeingpulledthroughtheringattherateof0.6ft/sec. How
fastistheboatapproachingthedockwhen13ftofropeareout?
9. Aballoonisataheightof50meters,andisrisingattheconstantrateof5m/sec.Abicyclist
passesbeneathit,travelinginastraightlineattheconstantspeedof10m/sec.Howfastis
thedistancebetweenthebicylistandtheballoonincreasing2secondslater?
10. Apyramid-shapedvathassquarecross-sectionandstandsonitstip. . Thedimensionsatthe
topare2m×2m,andthedepthis5m.Ifwaterisﬂowingintothevatat3m
3
/min,how
fast isthewaterlevelrisingwhenthedepthofwater(atthedeepestpoint) is4m? ? Note:
thevolumeofany“conical”shape(includingpyramids)is(1/3)(height)(areaofbase).
11. The e sun n is s rising at the rate of 1/4 4 deg/min, , and d appears to o be climbing into o the e sky
12. The e sun is s setting g at t the e rate of 1/4 deg/min, , and d appears to be e climbing g into the sky
13. Thetroughshowninﬁgure6.13isconstructedbyfasteningtogetherthreeslabsofwoodof
dimensions10ft×1ft,andthenattachingtheconstructiontoawoodenwallateachend.
6.2 RelatedRates
125
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Figure6.12
Sunset.
The angleθ θ was s originally30
, but t because of poor construction n thesides s are collapsing.
Thetroughisfullofwater. Atwhatrate(inft
3
/sec)isthewaterspillingoutoverthetop
ofthetroughifthesideshaveeachfallentoanangleof45
,andarecollapsingattherate
of1
persecond?
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Figure6.13
Trough.
14. Awoman5fttallwalksattherateof3.5ft/secawayfromastreetlightthatis12ftabove
the ground. . At t what rate is the tipofher shadow moving? ? At t what rateis her shadow
lengthening?
15. Aman1.8meterstallwalksattherateof1meterpersecondtowardastreetlightthatis4
16. Apolicehelicopterisﬂyingat150mphat t aconstant altitudeof0.5mile aboveastraight
milefromthehelicopter,andthatthisdistanceisdecreasingat190mph.Findthespeedof
thecar.
17. Apolicehelicopterisﬂyingat200kilometersperhourataconstantaltitudeof1kmabove
ofexactly2kilometersfromthehelicopter,andthatthisdistanceisdecreasingat250kph.
Findthespeedofthecar.
18. Alightshinesfromthetopofapole20mhigh. Aballis s falling10metersfromthepole,
126
Chapter6 Applications s oftheDerivative
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Figure 6.14
Fallingball.
(that
is, the e “north–south” ” road actually goes ina somewhat t northwesterly y directionfrom m P).
Recallthelawofcosines: c
2
=a
2
+b
2
−2abcosθ.
20. Doexample6.18assumingthatcarAis300metersnorthofP,carBis400meterseastofP,
bothcarsaregoingatconstantspeedtowardP,andthetwocarswillcollidein10seconds.
21. Doexample 6.18assumingthat 8seconds ago carAstartedfromrest at P P and d has s been
2
, and6 seconds s after car Astartedcar B
passedP movingeastatconstantspeed60m/sec.
200km/hrtotheeastofP atanaltitudeof2km,asdepictedinﬁgure6.15. Howfastisthe
distancebetweencarandairplanechanging?
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Figure6.15
Carandairplane.
200km/hrtotheeastofPatanaltitudeof2km,andthatitisgainingaltitudeat10km/hr.
Howfastisthedistancebetweencarandairplanechanging?
24. Alightshinesfromthetopofapole20mhigh. . Anobjectisdroppedfromthesameheight
fromapoint10maway,sothatitsheightattimetsecondsish(t)=20−9.8t
2
/2.Howfast