devexpress asp.net pdf viewer : Cut pages out of pdf online Library software component .net windows winforms mvc Calculus12-part742

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.
Cut pages out of pdf online - copy, paste, cut PDF pages in C#.net, ASP.NET, MVC, Ajax, WinForms, WPF
Easy to Use C# Code to Extract PDF Pages, Copy Pages from One PDF File and Paste into Others
extract pages from pdf online tool; cut pages out of pdf
Cut pages out of pdf online - VB.NET PDF Page Extract Library: copy, paste, cut PDF pages in vb.net, ASP.NET, MVC, Ajax, WinForms, WPF
Detailed VB.NET Guide for Extracting Pages from Microsoft PDF Doc
a pdf page cut; extract pages from pdf online
122
Chapter 6 6 Applicationsofthe e Derivative
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
..
.
.
.
.
.
..
.
.
.
.
.
..
.
.
.
.
..
.
.
.
.
..
.
.
.
..
.
.
.
..
.
.
.
..
.
.
..
.
.
..
.
.
..
.
.
..
.
.
..
.
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
..
..
.
..
..
..
.
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
...
..
..
..
..
..
...
..
..
...
..
..
...
..
...
..
...
...
..
...
...
...
...
..
....
...
...
...
...
....
...
....
....
....
....
....
....
.....
.....
.....
......
......
......
........
.........
...........
......................
................
......................
............
.........
.......
.......
......
......
.....
.....
....
.....
....
....
....
....
...
....
...
....
...
...
...
...
...
...
...
...
..
...
...
..
...
..
...
..
...
..
..
...
..
..
..
...
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
.
..
.
..
.
.
..
.
.
..
.
.
..
.
.
..
.
.
..
.
.
.
..
.
.
.
..
.
.
.
..
.
.
.
..
.
.
.
.
..
.
.
.
.
..
.
.
.
.
.
..
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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.)
LetRbe theradiusofthesphere,andlet randhbethebaseradiusandheightof
theconeinsidethesphere.Whatwewanttomaximizeisthevolumeofthecone:r
2
h=3.
HereRisaxedvalue,butrandhcanvary. Namely,wecouldchoosertobeaslargeas
possible|equaltoR|bytakingtheheightequaltoR;orwecouldmakethecone’sheight
hlargerattheexpenseofmakingralittlelessthanR. Seethecross-sectiondepictedin
C# HTML5 PDF Viewer SDK to view PDF document online in C#.NET
Image: Copy, Paste, Cut Image in Page. Link: Edit URL. Bookmark can view PDF document in single page or continue pages. Support to zoom in and zoom out PDF page.
acrobat remove pages from pdf; extract pages from pdf files
VB.NET PDF- View PDF Online with VB.NET HTML5 PDF Viewer
Remove Image from PDF Page. Image: Copy, Paste, Cut Image in can view PDF document in single page or continue pages. Support to zoom in and zoom out PDF page.
delete pages from pdf online; cut pages from pdf preview
6.1 Optimization
123
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
..
.
..
..
.
..
..
.
..
..
.
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
...
..
..
..
...
..
...
..
...
..
...
...
..
...
...
...
....
...
....
...
....
....
.....
.....
......
.......
.........
........................................
.........
.......
......
.....
....
.....
....
...
....
...
....
...
...
...
..
...
...
..
...
..
...
..
...
..
..
..
...
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
.
..
..
..
.
..
..
..
.
..
..
.
..
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
..
.
..
..
..
.
..
..
.
..
..
..
..
.
..
..
..
..
..
..
..
.
..
..
..
..
...
..
..
..
..
..
..
...
..
..
...
..
...
..
...
...
..
...
...
...
...
....
...
....
...
.....
....
.....
.....
......
........
..........
................................
..........
........
......
.....
.....
....
....
....
....
...
...
....
...
...
..
...
...
...
..
...
..
..
...
..
..
...
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
.
..
..
..
.
..
..
.
..
..
.
..
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
.
..
.
..
..
.
...
...
...
....
...
...
...
...
...
...
...
....
...
...
...
...
...
...
...
....
...
...
...
...
...
...
....
...
...
...
...
...
...
...
....
...
...
...
...
...
...
....
...
...
...
...
...
...
...
....
...
...
...
...
...
...
...
....
...
...
...
...
...
...
....
...
...
...
...
...
...
...
....
...
...
...
...
...
.....
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
..
.
..
.
..
...
...
...
...
...
...
....
...
...
...
...
...
...
...
....
...
...
...
...
...
...
...
....
...
...
...
...
...
...
....
...
...
...
...
...
...
...
....
...
...
...
...
...
...
....
...
...
...
...
...
...
...
....
...
...
...
...
...
...
...
....
...
...
...
...
...
...
....
...
...
...
...
...
...
...
....
...
...
...
.
(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%:
VB.NET Image: Image Cropping SDK to Cut Out Image, Picture and
and easy to use .NET solution for developers to crop / cut out image file This online tutorial page will illustrate the image cropping function from following
copy pages from pdf to new pdf; convert selected pages of pdf to word
VB.NET PDF Text Extract Library: extract text content from PDF
Extract highlighted text out of PDF document. Best VB.NET PDF text extraction SDK library and component for Online Visual Basic .NET class source code for quick
deleting pages from pdf file; extract one page from pdf reader
124
Chapter 6 6 Applicationsofthe e Derivative
EXAMPLE6.1.11
Youaremakingcylindricalcontainerstocontainagivenvolume.
Suppose that thetopandbottomare made ofa materialthat isN N timesasexpensive
(costperunitarea)asthematerialusedforthelateralsideofthecylinder. Find(interms
ofN)theratioofheighttobaseradiusofthecylinderthatminimizesthecostofmaking
thecontainers.
Letusrstchooseletterstorepresentvariousthings: hfortheheight,rforthebase
radius,V forthevolumeofthecylinder,andcforthecostperunitareaofthelateralside
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,
thereisnodierenceinthecostofmaterials,theheightistwicetheradius(ortheheight
isequaltothediameter).
.............................................................................................................
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
.
x
a x
A
D
B
C
b
Figure 6.1.5
Minimizingtraveltime.
C# PDF Text Extract Library: extract text content from PDF file in
Free online source code for extracting text from adobe Ability to extract highlighted text out of PDF C# example code for text extraction from all PDF pages.
copy one page of pdf to another pdf; copy page from pdf
VB.NET PDF - View PDF with WPF PDF Viewer for VB.NET
Image from PDF Page. Image: Copy, Paste, Cut Image in PDF pages extract, copy, paste, C#.NET rotate PDF pages, C#.NET Abilities to zoom in and zoom out PDF page.
copy web pages to pdf; cut pages out of pdf online
6.1 Optimization
125
EXAMPLE 6.1.12
Suppose youwant toreach a point Athat islocatedacrossthe
sandfromanearbyroad(seegure6.1.5).Supposethattheroadisstraight,andbisthe
distancefromAtotheclosestpointContheroad.Letvbeyourspeedontheroad,and
letw,which islessthanv,beyourspeedonthe sand. Rightnowyouareat t thepoint
D,whichisadistanceafromC. AtwhatpointBshouldyouturnotheroadandhead
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
C# WPF PDF Viewer SDK to view PDF document in C#.NET
Image from PDF Page. Image: Copy, Paste, Cut Image in PDF pages extract, copy, paste, C#.NET rotate PDF pages, C#.NET Abilities to zoom in and zoom out PDF page.
export pages from pdf preview; extract pages from pdf file
C# PDF Form Data fill-in Library: auto fill-in PDF form data in C#
Free online C# sample code can help users to fill in fill in form field in specified position of adobe PDF file. Able to fill out all PDF form field in C#.NET.
delete page from pdf file online; delete pages from pdf preview
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
alwayswanttocutacrossthesandwhenyouareadistancewb=
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
youranswers. )
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? )
VB.NET PDF- HTML5 PDF Viewer for VB.NET Project
Remove Image from PDF Page. Image: Copy, Paste, Cut Image in NET comment annotate PDF, VB.NET delete PDF pages, VB.NET PDF page and zoom in or zoom out PDF page
delete pages out of a pdf; delete blank pages from pdf file
VB.NET PDF - WPF PDF Viewer for VB.NET Program
Image from PDF Page. Image: Copy, Paste, Cut Image in Online Guide for Using RasterEdge WPF PDF Viewer to View PDF pages, zoom in or zoom out PDF pages and go to
deleting pages from pdf document; extract pages pdf
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). )
11. Findtheareaofthelargestrectanglethattsinsideasemicircleofradiusr(onesideofthe
rectangleisalongthediameterofthesemicircle). )
12. Foracylinderwithsurfacearea50,includingthetopandthebottom,ndtheratioofheight
tobaseradiusthatmaximizesthevolume. )
13. ForacylinderwithgivensurfaceareaS,includingthetopandthebottom,ndtheratioof
heighttobaseradiusthatmaximizesthevolume. )
14. Youwanttomakecylindricalcontainers s tohold1liter(1000cubiccentimeters)usingthe
leastamountofconstructionmaterial. Thesideismadefromarectangularpieceofmaterial,
andthis canbedonewithnomaterialwasted. . However,thetopandbottomarecutfrom
squaresofside 2r, so that 2(2r)
2
=8r
2
of materialis needed(rather than 2r
2
,whichis
thetotalareaofthetopandbottom). Findthedimensionsofthecontainerusingtheleast
amountofmaterial,andalsondtheratioofheighttoradiusforthiscontainer.)
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
thetopandbottom). Findtheoptimalratioofheighttoradius. )
16. Givenarightcircularcone,youputanupside-downconeinsideitsothatitsvertexisatthe
centerofthebaseofthelargerconeanditsbaseisparalleltothebaseofthelargercone.If
youchoosetheupside-downconetohavethelargest possiblevolume,whatfractionofthe
volume ofthelarger conedoesit occupy? ? (Let t H H andRbetheheightandbaseradiusof
thelargercone,andlethandrbetheheightandbaseradiusofthesmallercone. Hint: Use
similartrianglestogetanequationrelatinghandr.) )
17. Inexample6.1.12,whathappensifwv(i.e.,yourspeedonsandisatleastyourspeedon
theroad)?)
18. Acontainerholding g axedvolumeis beingmade intheshapeofacylinderwithahemi-
sphericaltop. (Thehemisphericaltophasthesameradiusasthecylinder.) ) Findtheratioof
heighttoradiusofthecylinderwhichminimizesthecostofthecontainerif(a)thecostper
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
cylindricallogofradiusr. )
.
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
..
..
.
..
..
..
..
.
..
..
..
..
..
..
..
...
..
..
..
...
..
...
..
...
...
...
...
....
....
....
.....
......
.........
......................
.........
......
.....
.....
...
....
...
...
...
...
...
..
...
..
..
...
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
.
..
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
..
.
..
..
..
.
..
..
..
..
..
..
.
..
...
..
..
..
..
..
...
..
...
..
...
...
...
...
...
....
....
.....
.....
.......
.................................
.......
.....
.....
....
....
...
...
...
...
...
..
...
..
...
..
..
..
..
..
...
..
.
..
..
..
..
..
..
.
..
..
..
.
..
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
.
..
.
..
.
...........................................................................................
.
..
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
...........................................................................................
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
..
.
..
"
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
topandbottomaremadeofamaterialthatistwiceasheavy(perunitarea)asthematerial
usedfortheside. )
28. Aconicalpapercupistohold1=4ofaliter. . Findtheheightandradius s oftheconewhich
minimizesthe amountof paper neededtomakethecup. . Use e the formular
p
r+hfor
theareaofthesideofacone.)
29. Aconicalpapercupistoholdaxedvolumeofwater.Findtheratioofheighttobaseradius
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?)
34. Youarestandingnearthesideofalargewadingpoolofuniformdepthwhenyouseeachild
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.)
..
..
..
..
..
...
..
..
...
..
...
...
...
...
....
....
.....
......
...........
.....
1
..
..
....
...
....
....
.....
......
..........
........
2
...
..
...
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
..
..
..
..
..
.
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
.
..
..
.
x
c x
a
b
Figure 6.1.7
Wadingpoolrescue.
Suppose we have twovariablesx and y (inmost problemsthe letterswillbe dierent,
but for r now let’s use e x and y) which are both changing g with h time. A A \related d rates"
problemisaprobleminwhichweknowoneoftheratesofchangeatagiveninstant|say,
130
Chapter 6 6 Applicationsofthe e Derivative
x_ =dx=dt|andwe e want tondthe otherrate e _y=dy=dtat t thatinstant. (Theuseof
x_ tomeandx=dtgoesbackto o Newtonand isstillusedforthispurpose, , especiallyby
physicists.)
Ifyiswrittenintermsofx,i.e.,y=f(x),thenthisiseasytodousingthechainrule:
y_=
dy
dt
=
dy
dx
dx
dt
=
dy
dx
x_:
Thatis,ndthederivativeoff(x),pluginthevalueofxattheinstantinquestion,and
multiplybythegivenvalueofx_=dx=dttogety_=dy=dt.
EXAMPLE6.2.1 Supposeanobjectismovingalongapathdescribedbyy=x
2
,that
is,it ismovingonaparabolicpath. . Ataparticulartime,sayt=5,thexcoordinateis
6andwemeasurethespeedatwhichthexcoordinateoftheobjectischangingandnd
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,
forexample x=f(y),orF(x;y)=k,orperhapsF(x;y)=G(x;y),whereF(x;y)and
G(x;y) are expressionsinvolving bothvariables. In n allcases, youcan solve the related
rates problem by taking g the derivative of both sides, , plugging g in all l the e known values
(namely,x,y,andx_),andthensolvingfory_.
Tosummarize,herearethestepsindoingarelatedratesproblem:
1. Decidewhatthetwovariablesare.
2. Findanequationrelatingthem.
3. Taked=dtofbothsides.
4. Pluginallknownvaluesattheinstantinquestion.
5. Solvefortheunknownrate.
EXAMPLE6.2.2
Aplaneis yingdirectlyawayfromyouat500mphatanaltitude
of3miles. Howfastistheplane’sdistancefromyouincreasingatthemomentwhenthe
planeis yingoverapointontheground4milesfromyou?
Toseewhat’sgoingon,werstdrawaschematicrepresentationofthesituation,as
ingure6.2.1.
Becausetheplaneisinlevel ightdirectlyawayfromyou,therateatwhichxchanges
isthespeedoftheplane,dx=dt=500. Thedistancebetweenyouandtheplaneisy;it
isdy=dtthatwewishtoknow. BythePythagoreanTheoremweknowthatx
2
+9=y
2
.
Documents you may be interested
Documents you may be interested