The feasibledomainof this functionis whichmakessensebecauseif is feet, then the
figurewill be two -foot-longfencesgoingawayfrom the housewith feet left for the width, Using
our First and SecondDerivativeTests and the methodfor findingabsolutemaximumsand minimumson a
closedinterval(in this problem, ), we see that the functionattainsan absolutemaximumat
at the point So the dimensionsof the pen shouldbe ; with thosedi-
mensions,the pen will enclosean area of
Recallin the LessonRelatedRatesthat we solvedproblemsthat involveda varietyof geometricshapes.
Let’s considera problemaboutsurfaceareasof cylinders.
Example3:
A certainbrandof lemonadesells its productin -ouncealuminumcans that hold ml
Find the dimensionsof the cylindricalcan that will use the leastamountof aluminum.
Solution:
We needto developthe formulafor the surfacearea of the can. This consistsof the top and bottomareas,
each and the surfacearea of the side, (treatingthe side as a rectangle,the lateralarea is (cir-
cumferenceof the top)x(height)).Hencethe primaryequationis
We observethat both our feasibledomainsrequire
In orderto generatethe secondaryequation,we note that the volumefor a circularcylinderis givenby
Usingthe giveninformationwe can find a relationshipbetween and. We
substitutethis valueinto the primaryequationto get
or