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Get Full Access to Calculus With Applications - 10 Edition - Chapter 4.3 - Problem 61a
Get Full Access to Calculus With Applications - 10 Edition - Chapter 4.3 - Problem 61a

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# Thermal Inversion When there is a thermal inversion layer

ISBN: 9780321749000 67

## Solution for problem 61A Chapter 4.3

Calculus with Applications | 10th Edition

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Calculus with Applications | 10th Edition

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Problem 61A

Thermal Inversion When there is a thermal inversion layer over a city (as happens often in Los Angeles), pollutants cannot rise vertically but are trapped below the layer and must disperse horizontally. Assume that a factory smokestack begins emitting a pollutant at 8 a.m. Assume that the pollutant disperses horizontally, forming a circle. If t represents the time (in hours) since the factory began emitting pollutants (t = 0 represents 8 a.m.), assume that the radius of the circle of pollution is r(t) = 2t miles. Let A(r) = ?r2 represent the area of a circle of radius r.a. Find and interpret A[r(t)].________________b. Find and interpret Dt A[r(t)] when t = 4.

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Exam2StudyGuide–Sections2.5,3.1-3.7,4.1-4.5 • Definitions o Set:acollectionofobjects o Element:anobjectinaset o EmptySet(alsoNullSet):asetwithnomembers,written∅or{} o FiniteSet:asetwithafinitenumberofelements o InfiniteSet:asetwithaninfinitenumberofelements o Cardinality:thenumberofelementsinaset,written|A| o UniversalSet:thesetofallelementsmentionedinagivencontext o Subset:asetwhoseelementsareallinanotherset,writtenA⊆B,“Aisasubset ofB.” o ProperSubset:IfA⊆BandA≠B,thenAisapropersubsetofB,writtenA⊂B o PowerSet:asetofallofthesubsetsofaset,writtenP(A),“ThepowersetofA” § IfA={1,2,3},P(A)={∅,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3} o Intersection:thesetofallelementssharedbytwosets,writtenA∩B,“the intersectionofAandB.”Intersectioncanapplytoinfinitesets. o Union:thesetofthecombinationofeveryelementineitherAorB,written A⋃B.Donotrepeatanelement. § {2,3}⋃{3,4}={2,3,4} o Difference:thesetofelementsinoneAandnotinB,writtenA–B o SymmetricDifference:thestofelementsinexactlyoftwosets,butnotinboth, writtenA⊕B=(A-B)⋃(B-A) o Complement:Allelementsintheuniversalsetthatarenotinaset,written = − o SetIdentity:anequationinvolvingsetsthatistrueregardlessofthecontentsof thesets o CartesianProduct:thesetofeveryorderedpairsuchthatthefirstentryisfrom AandthesecondisfromB,writtenAxB={(a,b):a∈Aandb∈B} § IfA={1,2}andB={a,b,c},AxB={(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)} o Strings:n-tuplesofcharactersdenotedwithoutparenthesesorcommas o Alphabet:thesetusedtocreatestringsviaCartesianproducts o BinaryString:astringwhosealphabetis{0,1} o EmptyString:astringwhoselength(k)is0,writtenλ o Concatenation:thejoiningoftwostrings § Ifs=010andt=10,st=01010.xλ=x o Disjoint:IfA∩B=∅,theyhavenoelementsincommon,theyaredisjoint. o PairwiseDisjoint:Inasequenceofsets , ,…, ,ifallcombinationsofsets ▯ ▯ ▯ aredisjoint,thesequenceispairwisedisjoint. o Partition:agroupofnon-emptysubsetsofAsuchthateachelementofAisin exactlyonesubset.Agroupofsets ▯ ,▯, is▯apartitionofAif: § ∀i(Ai⊆A) § ∀i(Ai≠∅) § ▯, ▯…, i▯pairwisedisjoint § A=▯ ▯ ▯ …⋃ ▯ o Function:somethingthatmapsonesettoanotherset,writtenf:XàY § f⊆(XxY) § foreveryx∈X,thereisexactlyoney∈Ysuchthat(x,y)∈f § Xisthedomainoff § Yisthetargetoff o ArrowDiagram:Avisualrepresentationofafunctionwithafinitedomain, connectingmembersofthedomaintomembersofthetargetwitharrows. o Range:Therangeoffis{y:(x,y)∈fforsomex∈X} o FloorFunction:floor(x)=⎣x⎦=thelargestintegerysuchthaty≤x o CeilingFunction:ceiling(x)=⎡x⎤=thesmallestintegerysuchthaty≥x o One-To-One(injective):eachelementoftherangeIsmappedtoonlyonce o Onto(surjective):thetarget=range,allelementsofthetargetareused) o Bijective:bothone-to-oneandonto o Bijection(One-To-OneCorrespondence):abijectivefunction o Composition:theprocessofapplyingafunctiontotheresultofafunction § Forf:XàYandg:YàZ,(gof)(x)=g(f(x)):XàZ § fogoh=fo(goh)=(fog)oh=f(g(h(x))) o Identityfunction:afunctionthatmapsasettoitselfandmapseveryelementto itself § TheidentityfunctionofA,A :AàAisdefinedasA (a)=aforalla∈A • Notes o Inaproofbycases,splittheproofintomultiplecases(i.e.xisevenandxisodd) o Whenexpressingtheelementsinaset,orderdoesn’tmatter.A={2,3,4}={3,2, 4} o Tosayanelementisinaset,usethenotationx∈A,“xisamemberofA.” o Usingasuperscript+or-nexttoasetname(i.e.Z )signifiesallthepositiveor negativemembers,respectively,ofaset. o Theelementsofasetcanbeothersets.A={∅,{2,3},Q} § Inthiscase,{2,3}∈A,but{2,3}⊄A,Because2∉Aand3∉A. o If|A|=n,|P(A)|=2 o Setoperatorscanecombinedlikelogicaloperators.Forexample,A⋃(B⋃C) o Todenotetheunionorintersectionofalargeumberofsets,writelikethis: ▯ § ▯▯▯ ▯ ∩ ▯ ∩ …∩▯ ▯ 2 § ▯ = …⋃ ▯▯▯ ▯ ▯ ▯ ▯ ▯ o |AxB|=|A|⋅|B| o Cartesianproductscanbemadeofmanysets.Forexample, ▯ ▯×…× = ▯ {k▯,▯,…, ▯ ∈▯ fo▯allisuchthat1 ≤ i ≤ n} o A =A×A×…×A,ktimes o Theinverseofafunction(f )switchesthetwovaluesoftheorderedpair -1 § Iff(1)=2,thenf (2)=1 § Afunctionhasaninverseonlyifitisone-to-oneandonto o Somefunctionsarebijectionsgivencertaindomainsandtargetsbutnotunder others o Iffunctionf:AàBhasaninversef :BàA,thenI =f of,I =fof A B • SetIdentities o CommutativeLaws § A∩B=B∩A § A⋃B=B⋃A o AssociativeLaws § (A∩B)∩C=A∩(B∩C) § (A⋃B)⋃C=A⋃(B⋃C) o DistributiveLaws § A∩(B⋃C)=(A∩B)⋃(A∩C) § A⋃(B∩C)=(A⋃B)∩(A⋃C) o IdentityLaws § A⋃∅=A § A∩U=A o ComplementLaws § A⋃=U § A∩ =∅ o DominationLaws § A⋃U=U § A∩∅=∅ o IdempotentLaws § A⋃A=A § A∩A=A o DoubleComplementLaw § = o DeMorgan’sLaws § ∩ = § ⋃ =∩ o AbsorptionLaws § A⋃(A∩B)=A § A∩(A⋃B)=A 3 o x∈(A∩B)↔(x∈A) (x∈B) o x∈(A⋃B)↔(x∈A) (x∈B) o x∈↔¬(x∈A) • ImportantMathematicalSets o Z:setofallintegers o N:setofallnatural(counting)numbers o R:setofallrealnumbers o Q:setofallrationalnumbers 4

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##### ISBN: 9780321749000

This full solution covers the following key subjects: circle, assume, inversion, represents, radius. This expansive textbook survival guide covers 34 chapters, and 2111 solutions. Since the solution to 61A from 4.3 chapter was answered, more than 478 students have viewed the full step-by-step answer. Calculus with Applications was written by and is associated to the ISBN: 9780321749000. The full step-by-step solution to problem: 61A from chapter: 4.3 was answered by , our top Calculus solution expert on 08/28/17, 03:31AM. The answer to “Thermal Inversion When there is a thermal inversion layer over a city (as happens often in Los Angeles), pollutants cannot rise vertically but are trapped below the layer and must disperse horizontally. Assume that a factory smokestack begins emitting a pollutant at 8 a.m. Assume that the pollutant disperses horizontally, forming a circle. If t represents the time (in hours) since the factory began emitting pollutants (t = 0 represents 8 a.m.), assume that the radius of the circle of pollution is r(t) = 2t miles. Let A(r) = ?r2 represent the area of a circle of radius r.a. Find and interpret A[r(t)].________________b. Find and interpret Dt A[r(t)] when t = 4.” is broken down into a number of easy to follow steps, and 112 words. This textbook survival guide was created for the textbook: Calculus with Applications , edition: 10.

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