MATHEMATICS SOLID GEOMETRY CE LICENSURE EXAMINATION PROBLEMS
6.
SOLID GEOMETRY
PRISM 1.
A circular circular cylinde cylinderr with with a volume volume o 6.54 m3 i! circum!cri"ed a"out a ri#ht $ri!m who!e "a!e i! an e%uilateral trian#le o !ide 1.25 m& 'hat i! the altitude o the cylinder in meter!( )*+, M -. /&++ 0&1, a. c. 0&0, 0&+, b. d.
7.
The "a!e! o a ri#ht $ri!m are he6a#on! with one !ide 6 cm lon#& I the volume volume o the $ri!m i! 450 cc7 how ar a$art are the "a!e!( )M++ M 54. ,&10 cm 0&55 cm a. c. /&,3 cm 0&45 cm b. d. 2ind the volume o a ri#ht circular cylinder who!e lateral area i! 25.918 m 2 and "a!e area o 7.068 m2& )M88 M /8. a. b.
5+&00 m/ 5,&3+ m/
c. d.
-8&,/ m/ 54&5- m/
.
A ri#ht re#ular he6a#onal $ri!m i! in!cri"ed in a ri#ht circular cylin cylinder der who!e hei#ht hei#ht i! 20 cm& The dieren dierence ce "etwee "etween n the circumerence o the circle and the $erimeter o the he6a#on i! 4 cm& Determine the volume o the $ri!m& )*88 M ,. +71,3 cc 5874,1 cc a. c. 55701, cc 587/31 cc b. d.
The "a!e! o a ri#ht $ri!m are $enta#on! with each !ide 6 cm lon#& The "a!e! are 12 cm a$art& 'hat i! the volume o the $ri!m in cm3( )*+1 M 4. 1//&15/&a. c. 10/&1-/&b. d.
!.
An o$en9to$ cylindri cylindrical cal tan: i! made o a metal !heet havin# an area o 43.82 m2& I the diameter i! 2 / 3 the hei#ht7 what i! the hei#ht o the tan:( )M85 M 3. /&-0 m 0&-/ m a. c. -&0/ m ,&-/ m b. d.
4.
A ri#ht $ri!m ha! he6a#onal "a!e! with an ed#e o 6 cm& The "a!e! are 12 cm a$art& 'hat i! the volume o the $ri!m in cm3( )*+1 M 54. 575-4&0 575-3&0 a. c. b. 575--&0 d. 575-0&0
1".
A trou#h ha! an o$en to$ 0.30 m "y 6 m and clo!ed vertical end! which are e%uilateral trian#le! 30 cm on each !ide& It i! illed with water to hal it! de$th& 2ind the volume o water in m3& )*85 M ,. 8&803 8&8/1 a. c. b. 8&8,4 d. 8&83,
5.
A trou#h ha! an o$en to$ 0.30 m "y 6 m and clo!ed vertical end! which are e%uilateral trian#le! 30 cm on each !ide& It i! illed with water to hal it! de$th& 2ind the volume o the water in m3& )M+4 M 54. 8&803 8&8/1 a. c. 8&8,4 8&83, b. d.
11.
A cylindrical tan: o$en at the to$ i! made o metal !heet havin# a total area o 49.48 m 2& I the hei#ht o the tan: i! 1.5 time! it! "a!e diameter7 determine the "a!e diameter o the tan:& )M8- M 53. /&, m /m a. c. -&, m /&- m b. d.
2.
A circular cylinder i! circum!cri"ed a"out a ri#ht $ri!m havin# a !%uare !%uare "a!e 1 m on an ed#e& The volume volume o the cylinder cylinder i! 6.283 m3& 2ind it! altitude in meter!& )M+3 M 1. , 3 a. c. 0&, 0 b. d.
3.
9999999 5 9999999
MATHEMATICS SOLID GEOMETRY TRUNCATED PRISM
1.
A cone wa! ormed "y rollin# a thin !heet o metal in the orm o a !ector o a circle 72 cm in diameter with a central an#le o 150 & 2ind the volume o the cone in cc& )M+4 M 5,. 171// 17100 a. c. 171-17155 b. d. °
12.
13.
A truncated $ri!m havin# a !%uare "a!e ha! a volume o 1,000 m3& The hei#ht o the $ri!m at each corner i! re!$ectively 7 m 7 7 m 7 10 m7 and 10 m& 'hat i! the area o the "a!e( )M85 M -. 551&3, m 5/0&/- m a. c. +-&5- m 5-,&-5 m b. d. A truncated $ri!m ha! a hori;ontal trian#ular "a!e ABC 7 AB = 10 cm7 BC = 12 cm 7 and CA = 8 cm& The vertical ed#e! throu#h A7 B7 and C are 20 cm7 12 cm7 and 18 cm lon#7 re!$ectively& Determine the volume o the $ri!m7 in cc& )M8/ M +. 335 34, a. c. ,,+ ,10 b. d.
CONE 14.
A ri#ht circular cone ha! a "a!e radiu! o x and an altitude o 3x& 'hat i! it! !lant hei#ht( )*+0 M 0. /&10-6 /&0306 a. c. /&53-6 -&4-46 b. d.
15.
'hat i! the hei#ht o a ri#ht circular cone havin# a !lant hei#ht o 10x and "a!e diameter o 2x( )M+, M --. /&/516 /6 a. c. /&536 +&+,6 b. d.
16.
A conical ve!!el ha! a hei#ht o 24 cm and a "a!e diameter o 12 cm& It hold! water to a de$th o 18 cm a"ove it! verte6& 2ind the volume o it! content in cc& )M+3 M /. /41&0 /4/&, a. c. /45&1 /4,&b. d.
17.
A cone wa! ormed "y rollin# a thin !heet o metal in the orm o a !ector o a circle 72 cm in diameter with a central an#le o 210 & 'hat i! the volume o the cone in cc( )*+1 M 51. 5/7/54 5/7153 a. c. b. 5/738d. 5/7,80
1!.
The !lant hei#ht o a ri#ht circular cone i! 5 m lon#& The "a!e diameter i! 6 m& 'hat i! the lateral area in m2( )*+4 M ,. /1&1 00 a. c. 01 08&4 b. d.
2".
A ri#ht circular cone ha! a "a!e diameter o e and an altitude o 2e& It! volume i!< )*++ M -+. / / πe =0 πe =a. c. / / πe =3 πe =/ b. d.
21.
The lateral area o a ri#ht circular cone o radiu! 4 c m i! 100.53 cm2& Determine the !lant hei#ht& )M88 M 0. 4 cm 3 cm a. c. + cm 58 cm b. d.
22.
The "a!e diameter o a cone i! 18 cm and it! a6i! i! inclined 60 with the "a!e& I the a6i! i! 20 cm lon#7 what i! the volume o the cone( )*88 M -+. 57,-0 cc 57-0, cc a. c. 5703+ cc 5734+ cc b. d.
23.
A conical ve!!el 1 m diameter at the to$ and 60 cm hi#h hold! !alt at a de$th o 36 cm rom the "ottom& How many cc o !alt doe! it contain( )*8- M +. /17-50 /,74+3 a. c. //7+-+ /57,10 b. d.
24.
The lateral area o a ri#ht circular cone i! 386 m2& I it! diameter i! one9hal it! altitude7 determine it! altitude in meter!& )*8/ M 58. -0&1 54&+ a. c. 51&0 --&, b. d.
°
9999999 - 9999999
°
MATHEMATICS SOLID GEOMETRY PYRAMID 25.
A re#ular trian#ular $yramid had an altitude o 9 m and a volume o 187.06 m3& 'hat i! the "a!e ed#e in m( )*+0 M 3. 54 5a. c. 50 53 b. d.
26.
The "a!e ed#e o a re#ular trian#ular $yramid i! 12 m& I the altitude i! 9 m7 what i! the volume in m3( )M+, M -0. 5+/&++ 541&83 a. c. 53+&10 548&5/ b. d.
27.
Determine the volume o a re#ular tetrahedron who!e !ide i! 3 m & )M85 M 5,. /&54- m / ,&/-5 m / a. c. / -&+4/ m 5&55+ m / b. d.
2.
The volume o a re#ular tetrahedron i! 3.182 m 3& 2ind it! total !urace area& )M8- M 51. 5-&01 m 53&41 m a. c. 5,&,+ m 58&0- m b. d.
2!.
How ar rom a verte6 i! the o$$o!ite ace o a tetrahedron i an ed#e i! 50 cm lon#( )M8/ M 4. /4&354 cm /+&0-5 cm a. c. 08&4-, cm 05&-50 cm b. d.
3".
The !urace area o a re#ular tetrahedron i! 173.2 cm2& 'hat i! it! altitude( )*8/ M 55. 4&- cm 1&- cm a. c. b. +&3 cm d. 3&, cm
32.
A ri#ht circular cone with an altitude o 9 m i! divided into two !e#ment!& One i! a !maller circular cone havin# the !ame verte6 with an altitude o 6 m& 2ind the ratio o the volume o the two cone!& )M+1 M 3. 5
33.
A ri#ht circular cone with an altitude o 8 cm i! divided into two !e#ment!& One i! a !maller circular cone havin# the !ame verte6 with volume e%ual to 1 / 4 o the "i##er cone& 2ind the altitude o the !maller cone& )M++ M 51. 0&,- cm ,&80 cm a. c. 3&10 cm 3&5- cm b. d.
34.
Two identical cone! with vertical a6i!7 one inverted and the other i! u$ri#ht ha! "a!e radiu! o 1.2 m and hei#ht o 4.8 m& Each cone contain! e%ual volume o oil havin# !$eciic #ravity o 0.8& I the de$th o oil in the inverted cone i! 2.4 m7 what i! the de$th o oil in the u$ri#ht cone( )M8- M 4. 8&3/ m 5&,0 m a. c. 8&01 m 8&-5 m b. d.
35.
A trian#ular $iece o land ha! one !ide mea!urin# 12 km& The land i! to "e divided into two e%ual area! "y a dividin# line $arallel to the #iven !ide& 'hat i! the len#th o the dividin# line( )M8- M -/. 3 1&3-/ a. c. 4&04, 4 b. d.
#RUSTUM 36.
SIMILAR #IGURES 31.
I the ed#e o a cu"e i! increa!ed "y 30%7 "y how much i! the !urace area increa!ed( )*+3 M 58. 31 > 3+ > a. c. 3/ > 3, > b. d.
9999999 / 9999999
The volume o a ru!tum o a re#ular trian#ular $yramid i! 135 m3& The lower "a!e i! an e%uilateral trian#le with an ed#e o 9 m & The u$$er "a!e i! 8 m a"ove the lower "a!e& 'hat i! the u$$er "a!e ed#e in m( )*+, M +. / , a. c. 0 3 b. d.
MATHEMATICS SOLID GEOMETRY 37.
3.
3!.
4".
41.
An artiicial la:e7 5 m dee$7 i! to "e du# in the orm o a ru!tum o an inverted $yramid& The level "ottom i! 8 m "y 80 m and it! to$ i! 10 m "y 100 m& How many cu"ic meter! o earth i! to "e removed( )M+1 M ,. 07831 07-43 a. c. 07051 073/3 b. d. The altitude o the ru!tum o a re#ular rectan#ular $yramid i! 5 m7 the volume i! 140 m3 and the u$$er "a!e i! 3 m "y 4 m & 'hat are the dimen!ion! o the lower "a!e in m( )*+1 M 5. 1&, 6 58 364 a. c. 0&, 6 3 + 6 58 b. d. The ru!tum o a re#ular trian#ular $yramid ha! e%uilateral trian#le! or it! "a!e!& The lower and u$$er "a!e ed#e! are 9 m and 3 m 7 re!$ectively& I the volume i! 118.2 m 37 how ar a$art are the "a!e!( )*+4 M 58. +m 1m a. c. 4m 58 m b. d. The ru!tum o a re#ular trian#ular $yramid ha! e%uilateral trian#le! or it! "a!e! and ha! an altitude o 8 m & The lower "a!e ed#e i! 9 m & I the volume i! 135 m37 what i! the u$$er "a!e ed#e( )M88 M 50. -m 0m a. c. ,m /m b. d. A lateral ed#e o the ru!tum o a re#ular $yramid i! 1.8 m lon#& The u$$er "a!e i! a !%uare 1 m x 1 m and the lower "a!e 2.4 m x 2.4 m !%uare& Determine the volume o the ru!tum in m3& )*8- M 1. 0&3 ,&1 a. c. /&/ 3&, b. d.
PRISMOID 42.
The "a!e diameter o a certain !olid i! 40 cm& 2ind the volume o the !olid in cc i every cuttin# $lane $er$endicular to a #iven diameter i! a !%uare& )M++ M 5+. ,-7331 cc ,-7//0 cc a. c. /-7,31 cc 0-7331 cc b. d.
43.
A !olid ha! a circular "a!e o "a!e radiu! 20 cm& 2ind the volume o the !olid i every $lane !ection $er$endicular to a certain diameter i! an i!o!cele! ri#ht trian#le with one le# in the $lane o the "a!e& )*88 M 3. -57/// cc 547331 cc a. c. -075,, cc -870// cc b. d.
44.
The "a!e radiu! o a certain !olid i! 20 cm& 2ind the volume o the !olid in cc i every cuttin# $lane $er$endicular to a #iven diameter i! an e%uilateral trian#le& )*85 M -8. 537,0/&-+ 54701,&-5 a. c. 5+70,/&41 507-5/&+4 b. d.
$OLUME O# INTERSECTIONS 45.
A !%uare hole 2” x 2” i! cut throu#h a 6-inch diameter lo# alon# it! diameter and $er$endicular to it! a6i!& 2ind the volume o wood that wa! removed& )*++ M 55. -1&/- in/ -5&14 in/ a. c. b. -/&,0 in/ d. /0&3- in/
46.
A cylinder o radiu! 6 m ha! it! a6i! alon# the X-a6i!& A !econd cylinder o the !ame radiu! ha! it! a6i! alon# the Y-a6i!& 2ind the volume7 in the ir!t octant7 common to the two cylinder!& )M88 M 5,. 433 m / 500 m / a. c. / / 575,- m -44 m b. d.
9999999 0 9999999
MATHEMATICS SOLID GEOMETRY SP%ERE 47.
The volume o !$here o radiu! 1.2 m i!< )M+0 M 3. 37333 m / 47,31 m / a. c. 075,3 m / 17-/4 m / b. d.
4.
The corner! o a cu"ical "loc: touch the clo!ed !$herical !hell that enclo!e! it& The volume o the "o6 i! 2,744 cc& 'hat volume in cc in!ide the !hell i! not occu$ied "y the "loc:( )M+1 M 51. 170-5 071-5 a. c. -7105 57/-5 b. d.
4!.
5".
'hat i! the !urace area o a !$here who!e volume i! 36 m 3( )*+4 M -8. ,-&1 m03&3 ma. c. 04&1 m ,0&3 mb. d.
53.
A mi6ture com$ound rom e%ual $art! o two li%uid!7 one white and the other "lac:7 wa! $laced in a hemi!$herical "owl& The total de$th o the two li%uid! i! 6”& Ater !tandin# or a !hort time7 the mi6ture !e$arated& The white li%uid !ettled "elow the "lac:& I the thic:ne!! o the !e#ment o the "lac: li%uid i! 2”7 ind the radiu! o the "owl in inche!& )M+3 M --. 1&,/ 1&1/ a. c. 1&// 1&+/ b. d.
54.
The volume o the water in a !$herical tan: i! 1,470.265 cm3& Determine the de$th o water i the tan: ha! a diameter o 30 cm& )*+3 M 55. 4 3 a. c. 0 58 b. d.
55.
The volume o water in a !$herical tan: havin# a diameter o 4 m i! 5.236 m 3& Determine the de$th o the water in the tan:& )*+3 M 5,. 5&8 5&0 a. c. 5&5&3 b. d.
56.
20.5 m3 o water i! in!ide a !$herical tan: who!e radiu! i! 2 m&
The volume o a !$here i! 52 m3& Determine it! !urace area in !%uare meter!& )*++ M -0. ,0&5 ,3&/a. c. +/&/ 31&/, b. d.
2ind the hei#ht o the water !urace a"ove the "ottom o the tan:7 in meter!& )M+4 M 53. -&1 -&/ a. c. -&, -&5 b. d.
SP%ERICAL SEGMENT 51.
52.
'hat i! the area7 in m27 o the ;one o a !$herical !e#ment havin# a volume o 1,470.265 m3 i the diameter o the !$here i! 30 m( )*+0 M ,. a. 3,+&1/0 c. 404&-/ ,3,&041 1,/&+4b. d.
57.
A !$here havin# a diameter o 30 cm i! cut into 2 !e#ment!& The altitude o the ir!t !e#ment i! 6 cm& 'hat i! the ratio o the area o the ;one o the !econd !e#ment to that o the ir!t( )M+, M -/. 0&,<5 ,<5 a. c. ,&,<5 0<5 b. d.
9999999 , 9999999
A mi6ture com$ound rom e%ual $art! o two li%uid!7 one white and the other "lac:7 wa! $laced in a hemi!$herical "owl& The total de$th o the two li%uid! i! 6”& Ater !tandin# or a !hort time7 the mi6ture !e$arated& The white li%uid !ettled "elow the "lac:& I the thic:ne!! o the !e#ment o the "lac: li%uid i! 2”7 ind the radiu! o the "owl in inche!& )*++ M 58. 1&,/ 1&1/ a. c. 1&// 1&+/ b. d.
MATHEMATICS SOLID GEOMETRY SP%ERICAL &EDGE 5.
2ind the radiu! o the !$herical wed#e who!e volume i! 12 m3 with a central an#le o 1.8 radian!& )M8/ M 1. -&/3 m -&,- m a. c. -&1/ m -&5, m b. d.
5!.
Determine the volume o a !$herical wed#e o radiu! 2 m and a central an#le o 1.25 radian!& )*8/ M +. 3&31 m/ +&4, m/ a. c. 4&30 m/ ,&10 m/ b. d.
9999999 3 9999999
