We startedwith the generalvolumeformula , but quicklyrealizedthat we did not havesufficient
informationto find sincewe had no informationaboutthe radiuswhenthe waterlevel was at a particular
height.So we neededto employsomeindirectreasoningto find a relationshipbetweenr and h,
. We then madean appropriatesubstitutionin the originalformula ) and
wereable to find the solution.
We startedwith a primaryequation, , that involvedtwo variablesand provideda general
modelof the situation.However, in orderto solvethe problem,we neededto generatea secondary
equation, , that we then substitutedinto the primaryequation.We will face this samesituation
in mostoptimizationproblems.
Let’s illustratethe situationwith an example.
Example2:
Supposethat Marywishesto makean outdoorrectangularpen for her pet chihuahua.She wouldlike the
pen to enclosean area in her backyardwith one of the sidesof the rectanglemadeby the side of Mary's
houseas indicatedin the followingfigure.If she has ft of fencingto workwith,whatdimensionsof the
pen will resultin the maximumarea?