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Theta Three-Dimensional Geometry 2013 ΜΑΘ National Convention Note:Forallquestions,answer“(E)NOTA”meansnoneoftheaboveanswersiscorrect. 1. Whatisthesquareofthelengthofthelongestdiagonalofacubehavingedgelength3? (B)9√3 (C)27 (D)27√3 (E)NOTA (A)3√3 2. Asolidspherewithradiuslength17iscutbyaplanesothattheintersectionofthe sphereandtheplanecreatesacirclewithareaof225 .Findthedistancebetweenthe planeandthegreatcircleofthespherethatisparalleltotheplane. (A)12 (B)8 (C)15 (D)10 (E)NOTA 3. Atankintheshapeofarightrectangularprismhasabasewithdimensionsof5meters by10metersandaheightof14meters.Thistankisfilleduphalfwaywithwater.A metalconethatweighs357gramsandhasadensityof3gramspercubicmeteris droppedintothetank,causingthewaterleveltogoup.Whatisthepositivedifference, inmeters,intheheightofthewaterbeforeandaftertheconewasdroppedintothe tank? (A)2.38 (B)2.48 (C)2.66 (D)2.76 (E)NOTA 4. Afrustumhasparallelcircularbaseswithareas64 and16 .Theheightofthefrustum is36.Findthevolumeofthefrustum. (A)1776 (B)1632 (C)1488 (D)1344 (E)NOTA 5. Asolidcubehasanedgelengthof16centimeters.Atriangleis drawnwithverticesatthemidpointsofthreeconnectingedges, andone“corner”ofthecubeistruncated(i.e.,cutwithaplanar knife)sothatthistriangleisshowing.Theremainingfigureisa solidwith7faces.Find,insquarecentimeters,thesurfaceareaof thissolid.(Note:Diagramtotherightisnotnecessarilytoscale.) (A)1440+32√3 (B)1440+64√3 (C)1184+32√3 (D)1440+64√3 (E)NOTA 6. Thecrosssectionofametalcylindricalpipeconsistsoftwoconcentriccircleswith constantradiusthroughoutlengthofthepipe.Iftheareaoftheannulusis160 andthe thicknessofthepipeis4,findthelengthofthediameterofthepipe. (A)11 (B)22 (C)44 (D)66 (E)NOTA Page 1 of 5 Theta Three-Dimensional Geometry 2013 ΜΑΘ National Convention 7. Acubohemioctahedronisapolyhedronthathashowmanyfaces,edges,andvertices? (A)Numberoffaces=10;Numberofedges=24;Numberofvertices=12. (B)Numberoffaces=10;Numberofedges=20;Numberofvertices=12. (C)Numberoffaces=14;Numberofedges=24;Numberofvertices=12. (D)Numberoffaces=10;Numberofedges=24;Numberofvertices=16. (E)NOTA 8. LetAandBbeoppositeverticesofaunitcube(i.e.,thedistancebetweenAandBis√3). Findtheradiusofasphere,whosecenterisintheinteriorofthecube,thatistangentto thethreefacesthatmeetatAandalsotangenttothethreeedgesthatmeetatB. (A)3 √5 √ (B) (C)2 √2 (D) √ (E)NOTA 9. Joewantstomeltasolidmetalspherewithradiuslength12ft.intoarightcircular cylinderwithabasehavingareaof36 ft².Howmanyfeettallwillthecylinderbe? (A)16 (B)24 (C)32 (D)64 (E)NOTA 10. Jillhascongruent,perfectlysphericaloranges.Shearranges theminapyramid‐likestructureonaleveltablewherethe baseismadeupof3x3oforangesthataretangenttoeach other,thesecondlayerismadeupof2x2oforangesthatare alsotangenttoeachother,andthetoplayerhas1orange. Eachorangeinthetoptwolayersisplacedsothatitis tangenttofourorangesinthelayerbelow.Thecross‐ sectionoftheresultingstructurethatincludesthegreat circlesof6orangeslookslikethefiguretotheright.Howmanyorangesareinthe structure? (A)12 (B)13 (C)14 (D)15 (E)NOTA 11. WithreferencetoProblem#10,themaximumheightofthispyramid‐likestructureis4. Whatisthelengthoftheradiusofeachindividualorange? (B) (C) (D) (E)NOTA (A) √ √ √ Page 2 of 5 √ Theta Three-Dimensional Geometry 2013 ΜΑΘ National Convention 12. Acubewithsidelength7isdippedinabucketofpaint.Thiscubeisthencutinto343 unitcubes.Howmanycubeshaveexactlyonesidepainted? (A)125 (B)135 (C)140 (D)150 (E)NOTA 13. Whatisthevolumeofahexagonalpyramidwithbasesidelengthof6andheightof3? Thebaseofthepyramidofaregularhexagon. (B)54√3 (C)27√3 (D)9√3 (E)NOTA (A)108√3 14. Findthedistancebetweenthepoints(17,‐73,87)and(53,4,2). (A)5√574 (B)120 (C)85√2 (D)95√2 (E)NOTA 15. Whatisthetotalsurfaceareaofacubewithedgelengthequaltotheradiusofasphere withvolume288 ? (A)72 (B)144 (C)216 (D)288 (E)NOTA 16. Asphereofradius6centimetersismeltedintoacylindricalwirewithaconstantradius of0.06centimeters.Whatisthelengthofthewire,inmeters? (A)80000 (B)8000 (C)800 (D)80 (E)NOTA 17. Joewantstofitachopstickinacylindricalcan.Theradiusofthecanhaslengthm/2, andtheheightofthecanisequaltotheslantheightofaconewithradiuslengthm/2 andheight2m.Ifthewidthofthechopstickisnegligible,find,intermsofm,thelength ofthechopstickthatwillexactlyfitinthecan. (B)m√21/3 (C)2m√21 (D)2m√21/3 (E)NOTA (A)m√21/2 18. Bettycutsacircularpizzainto slices,forming circularsectors.Themeasuresofthe centralanglesoftheseslices,whenarrangedfromsmallesttolargest,forman arithmeticprogression.Ifthesmallestslicehasananglemeasureof3°andthelargest hasananglemeasureof37°,howmanyslicesdidBettycut? (A)14 (B)15 (C)16 (D)17 (E)NOTA 19. Arightcircularconeiscutbyaplaneparalleltothebase,splittingtheconeintoa smallerconeandafrustum.Thefrustumhasbaseswithdiameterlengthsof6and30. Theheightofthefrustumis12.Whatistheslantheightofthesmallercone? (A)3 (B)3√2 (C)5/2 (D)5√2/2 (E)NOTA Page 3 of 5 Theta Three-Dimensional Geometry 2013 ΜΑΘ National Convention 20. Ignoringunits,findthesumofthelateralsurfaceareaandthevolumeofthefrustumin Problem#19. (B) (1116+216√2) (A) (1116+288√2) (C) (837+288√2) (D) (837+216√2) (E)NOTA 21. Theratiobetweentheradiioftwospheresis5:3.Whatistheratioofthevolumeofthe largerspheretothevolumeofthesmallersphere? (A) 25√2/3 (B)25/9 (C)125√2/9 (D)125/27 (E)NOTA 22. Twosphericalballslieonthegroundtangenttoeachother.Theradiusofoneballis18 feetandtheradiusoftheotheris24feet.Whatistheheight,infeet,fromthegroundto thepointoftangencyofthetwospheres? (A)108/7 (B)144/7 (C)216/7 (D)288/7 (E)NOTA 23. Whatisthetotalsurfaceareaofapyramidwithasquarebasewherethesideofthe baseis10andtheheightofthepyramidis12? (A)300 (B)330 (C)345 (D)360 (E)NOTA 24. Alightbulbisintheshapeofacube.Ithasanedgelengthofsixcentimetersandcan onlyilluminateamaximumdistanceofthreecentimetersfromanypointofthecube. Whatisthemaximumtotalspace,incubiccentimeters,thatthelightbulbcanlightup thatdoesnotincludetheinsideofthecube? (A)648+198 (B)648+214 (C)648+368 (D)648+228 (E)NOTA 25. Whatistheslantheightofarightcircularconewhosebasehasacircumferenceof170 andheightof132? (A) 157 (B)2√6117 (C)11√203 (D)9√327 (E)NOTA 26. Whatisthevolumeofaregulartetrahedronwithsidelength4? (A)4√2/3 (B)8√2/3 (C)4√3/3 (D)8√3/3 (E)NOTA 27. Whatisthesurfaceareaofaregularicosahedronwithsidelength12? (B)720√3 (C)810√3 (D)1080√3 (E)NOTA (A)540√3 Page 4 of 5 Theta Three-Dimensional Geometry 2013 ΜΑΘ National Convention 28. Kellywantstopaintthewallsandtheroofofherroom,whichisintheshapeofaright rectangularprism.Herroomis18feetby21feet,andis12feettall.Whatisthesurface areasheneedstocover,insquareyards? (A)146 (B)164 (C)156 (D)165 (E)NOTA 29. Asphereisinscribedinatetrahedronwithverticesat 4,0,0 , 0,3,0 , 0,0,1 ,and 0,0,0 .Thevolumeofthespherecanbeexpressedintheform / ,where and arerelativelyprimepositiveintegers.Find . (A) 141 (B)139 (C)137 (D)135 (E)NOTA 30. Asealedcontainerisintheshapeofarightcircularconethatis4inchestallandthe baseoftheconehasaradiusof3inches.Thereiswaterinthecontainer,andwhenthe containerisheldwithitsvertexdownandbasehorizontal,thewaterlevelis2inches awayfromthevertex.Whenthecontainerisheldwithitsvertexupandbase horizontal,thewaterlevelis inchesawayfromthevertex.Findthevalueof . (B)2 ⋅ √5 (C)2 (D)2 ⋅ √7 (E)NOTA (A)2 ⋅ √3 Page 5 of 5