数学建模(西点军校资料)C1

《数学建模(西点军校资料)C1》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。

1、EmergencyDeploymenttoPanamaJerryKobylskiIntroductionYourbattalionhasrecentlybeendeployedtoPanamainresponsetoanimmigrationcrisis.Becauseyouknowyouwillbethereforseveralmonthsasaresponseforceintheeventofanuprising,yourtheatercommanderhasauthorizedfundsfortheconstructionofabasecamp.Astheassistantbatt

2、alionS3,youarechargedwithseveralmissionswhichinvolvethepreparationofthecampforyourbattalion.Mostofthesemissionsinvolvemathematicalmodeling.Example1:Youneedtoconstructagraywaterpitforthebattalion.Agraywaterpitisalargerectangularholewherewaterfromshowerscollectsothatthegroundcanabsorbthewater.Youmu

3、stconstructanopentoprectangularpitwithasquarebaseofxfeetoneachsideandaheightofyfeet.Thepitwillbeplacedsothatitcancatchalloftherunoffwaterfromtheshowers.Itmustbelargeenoughtosupportthepeakshowertime(rightafterPT)whichisforonehour.Twentyshowerheadsusuallyrunduringthishour,eachreleasingabout2cu.ftpe

4、rminute.Considertheabsorptionrateofthegroundtobenegligible.Thecostsassociatedwiththeconstructionofthepitinvolvenotonlythematerial,butalsoanexcavationcharge.Thetotalcostcanbewrittenasc(x,y)=25(x+4xy)+10xy.SincethebattalionmustcontractwiththelocalPanamanianengineersforthisproject,yourcommanderhasdi

5、rectedyoutofindthedimensionsofthepitthatwillminimizethecost.Whatare25thesedimensionsandwhatisthecost?AssumptionsWemustassumethefollowing:theabsorptionofthewaterbythegroundwillbenegligible;allofthewaterfromtheshowerheadswillendupinthepit;theshowerheadsruncontinuouslyandthattherateofeachisexactly2c

6、u.ft.perminute;andthegivencostfunctionisanaccuraterepresentationoftheassociatedcosts.Variables:x:thelengthandwidthofthebottomofthepitinfeety:theheightofthepitinfeetc(x,y):thetotalcostoftheconstructionofthepitindollarsSincethebaseofthepitwillbeasquare,weknowthatthevolumewillbev=2xy.Thisvolumemustb

7、eabletosupporttheamountofwatercomingfromtheshowers.Thus,32ftV=xy=20(2)´60min=2400min2400y=2xNowwecansubstitutethisinforyinthecostequation:22c(x,y)=5x+20xy+10xy=5x+30xy2400720022c(x)=5x+30x()=5x+2xxDifferentiatingc(x)gi