Inthisvideo,I'llworkthroughsomeproblems.Sofirstlet'ssaywewereaskedtofindequationsforthespherecenteredatnegative1,0,2,andradiussquareroot7.Sorememberthatingeneral,theequationforaspherecenteredatX0,Y0,Z0,andwithradiusrisxminusX0squaredplusmyYminusY0squaredplusZminusZ0squaredequalsrsquared.Soinourcase,thatbecomesXplus1squaredplusYsquaredorplusZminus2squaredequals7.IfyoulikeyoucouldexpandthisoutasXsquaredplus2xplus1plusYsquaredplusZsquaredminus4Zplus4equals7.AndthenrewritethisasXsquaredplus2XplusYsquaredplusZsquaredminus4Zisequalto2.Butformostpurposes,it'llbealoteasiertojustleaveitinthisform.SotheonlyreasonIdidthatexpansionistoprepareforthenextproblem,whichis--sonowlet'ssayweweregiventheequationforasphereinthatexpandedoutform.Nowwewanttofindthecenterandradiusofthesphere,andthenitasksustoalsofindthetwopointsonthespherethathavethehighestandthelowestz-coordinate.Soforparta,we'llcompletethesquare.Soxplus2squaredisxsquaredplus4xplus4y,yminus1squaredisysquaredminus2yplus1,andzminus3squarediszsquaredminus6zplus9.Sotocompletethesquare,weneedtoadd14tobothsidesoftheequation.Sothenwegetxplus2squaredplusyminus1squaredpluszminus3squaredequals25,whichofcourseis5squared.SothecenterisCequalsnegative2,1,3,andtheradiusisaequalsto5Thenforpartb,soifwehaveasphereandlet'ssaythisisthecenterpoint,hascenteratx0,y0,z0,andwecanseethatthepointswiththehighestandlowestsetcoordinatesaregoingtobethepointsdirectlyaboveanddirectlybelowthecenter.Sincethishasradiusr,thesearegoingtohavecoordinatesx0,y0,z0plustheradiusandx0,y0z0minustheradius.So,nowthepointouranswerispointswiththehighestandrespectivelylowestz-coordinatesarenegative2,1,8orlet'ssayrespectivelynegative2,1,negative2.