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ID:43492319
大小:182.28 KB
页数:6页
时间:2019-10-08
《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
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