Download Theta Three-Dimensional Geometry 2013 ΜΑΘ

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
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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)
√
√
√
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√
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
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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
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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
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