11-4 Volume of Prisms and Cylinders 'olume is the space that a three-dimensional figure occupies. Since it is 3-D, it is measured in cubic units. Cavalieri's Principle - f two 3-D figures have the same height and the same cross-sectional area at every level, then they have the same volume. The following prisms have the same height. Since the area of each cross section is 6 in2, by Cavalieri's Principle, their volumes will be the same. Base is a 3 in by,2 in rectangle Base is (1 ~ pcrclleloqrcm ' 3' n The volume of a prism or cylinder is the product of the base area and the height. V -:::.~ ~ Find the volume of each of the following prisms or cylinders. Give exact answers and onswers rounded to the necrest tenth. "Y:0~O /_n h 11.<, 1'1 (j;. l ('d,\.\-)l\~} ";l. 5-:: 1 :\2 2. 'J~ \tp'\\'(\'l) :. \ q d.. '\\ c..'m; 3. 4. 10 15
5. The volume of '~in-der with height 8 in is 200 TT in 3 Find the length of the radius. \[~e,+ ~ob}f ::.'j('('1..(8) \ V':. 5~J 6. Find the volume of the right prism. (note: all anglesare right angles) ~:: 'i (1') -t?>o (~\) :: ag +- to ~O =- (p 58 \j :: ~ 't;. {P 53(5) ~ ::. 2>J. 10 la 21 30 7. The plane region is revolved completely about the v-cxis. Describe the sold and find its.volumein ter,!,s of "!. ~ ~'f\in~ef ('(;.'1, ~::~5)~ a. ~ ty'':\j5j ~~ ~\~Ci\' V=- \(o'\\(5j=- go'll ~ \-; \'tt C'1 \ ie, 'if C 5) 2> B:: otl-'[ ( V c, 80 'if - 5 '\\ z: 75 'if V- ::::,'/5"11'" ~~ 8. A cylindrical "hole" (with diameter 6 cm) has been cut out of a prism. Find the volume of the remaining solid. b ::. l<ec\- - 0 :: \ 2. (\5) -- q 7f 1// zz,, ~ $ o,, ~,..---..~~ 22 em 'B - \<60 -- q '\\ \[::. (\80 - q'\\) ~~ :.:(3q (00- \ q fi: 1\') <:VV\3, (' D... 15l\'l.) (z-.;!.):: 3<1 (00 'i 0-'1" { f\ ~ f'(\ - '\l 0 c c.; \. = ~ \\ (Z"2-; =- \ q g'i\ \j 0 {- So \\ ~ = 39 lo0 - \ q~ '\\
11-~ Volume of Pyramids and Cones The volume of a pyramid is one third the product of the area of the base and the height of the pyramid. V = 1 -Bh 3 Because of Cavalieri's Principle, the volume formula is true for all pyramids, including oblique pyramids. The height of an oblique pyramid is the length of the perpendicular segment from the vertex to the plane of the base. The volume of a cone is one third the product of the area of the base and the height of the cone. V = ~Bh, or V = ~"r2.1c This formula applies to all cones, Jluding oblique cones. Examples: 1. Find the volume of each of the following solids. /1 to~ 7cm b:: \.\C\ \j~ 4Q(\O) ~ d \ ~ ~ -.: '!O ~ "3 C;M 2. B=-8\1 \j -:.~lf (fl.) ------ 3.,22 in. ~~ ~(\fs)cl-l.,) ~ \C\$3,,=- \ C\ ~ ( \ ~) 3 ~ -:. ~ 5 <is'.-vtv 4. 'b -:.q 'T\ Ci\ = C1'\\ (1) 7 rn :. ~3'\\ '" ~ GoV\e:. '11\ (4: : \~\\ro~ ~ r':-!' /W S":N 3 So\\6. ~ t> ~ \\ + \2 \\ ~'1 \\N' 5. "l.. '1-' 2-1?> -=- \ 'l -:.\4'-\ -C--\ ~ 1. -t -{ "2.. -:.- ~ '?(\SM~ \l)~(\~) i1..-:.~\'(-3co-:'~u? 12 ft '; \ '1:t~ ~ ~ 't '!.- ~ 11 12ft P"\V' - ~u, l~-r1): 9("fttlr~ 'b So\leA - ~:l 'l? 4- q ("il) \-\-j \2
1 6. 2 ft 7. Find the radius of a circular cone whose volume is B~: an~~ght;s 6'1" \""- a';"" 6 ft ~--qrrr \/ -=- 9 '1f (2,) ~ k>'\t f\ ~ 3 8. Find x if the volume is 126 cm2. ~:: ~ (a.')l'1-) :: \]-- ~ ~ 'd-lp -=- ~.5 'X-. ~) ~ 318 -=- lo ~ 'X- Co t-vv) =. ~ i ; 'i\:~~ ~ - S- \ 2'\\ -: 1\ Y' "2... lo 3 2- ~41\ ~ (o't\(' 4 -:..'('''2-9. Find the volume of a circular cone whose radius is 12 ft and whose surface area is 300TT ft2. SA:. LA 'r B 300)1-= ~(g,4)f)~.\. \4-41 3Db ~ ';)..Q ~\ \ l.ftt \5~ -=- \ ~.9. _ \ L\l\ \\ (5) \ ~ -=- i 'J... ---;- ),J= :A40'lr~~ j 10. A 120 0 sector is cut from a circle with a radius of 9 in. The 120 0 sector is "r~lled" to create a cone. Find t~~;i:?~.*-~~.o~f.the resulting cone.?j'z."'" -R 1. ':. q~,';lllilillil1jrmi, cw:.&n"tj - ~ (\~1) ~,'"g, -q ~1 2 lllli~lltljf_~ ~~o ~ ~ q~ (~~) ~if{2d C1 ~:;. 3(81\i'):J; L.f\:: irk o;\d..i ~~~~ 0Jte-CL. '\ -:..3 ~ c ~ 11 11. The plane region is revolved completely about the given line to sweep out a solid of revolution. Describe the solid and then ~ind its volume in terms of TT.. \ y (a) about the x-cxrs (b) about the y-axs 1 1 Cffi'tL c :.~ ~:. ~ 'J ~ ~ rrr (:J 3 V~ \d-'"kj ~ wi;:;tfu ~ CMt cnct l "/. ~ _... '(":- \.lc ~:= 0 " ';1/ X ~ = llol\(?) '"'+~~ ", ~ ~ llc1\c~)-=- \61T1?:> 1.3 tsd\\~ ~ L\8 ~ - \\.o'\l :: ~d.ttr u y-- \ )
11-6 Surface Areas and Volumes of Spheres A ~e 1('((,..: is the set of all points in space equidistant from a given point called the e.ey\.te<". A (0..0\ \ V':> is a segment that has one endpoint at the center and the other endpoint on the sphere. A d\awl ejec is a segment passing through the center with endpoints on the sphere. When a plane and a sphere intersect in more than one point. the intersejion is a <,-'V'C \e. f the center of the circle is also the center of the sphere, the circle is called a ~V"eo...t t\fc\.j. The circumference of a great circle is the C \( c..v M~e.\{'eV\c.e of the sphere. A great circle divides a sphere into two bf.m~s p he 'f es Surface Area of a Sphere: Volume of a Sphere: Example 1: Find the surface area of a sphere whose diameter is 18 ft. S A ~ 4'1\ lq,>'2-- -=- ~ ~ '-\ \\ -\--\-1- f::. q Example 2: Find the surface area of a sphere where one of its great circles has an area of 3611" in2. 'f:. t, Example 3: The circumference of a rubber ball is 13 cm. Find its surface area to the nearest whole number. C. -=- ~f\\x' :.. \ ~ S l. A::..Lt'T\r f\\v'--lo.5 ~ -:. '-\-1l: l2.ol') r r;.- ~.01 ~53,8't5~.. SA:: 5t..t ~V;- Example 4: Find the volume of the sphere whose diameter is 30 cm. r:.\«::5
Example 5: The volume of a sphere is 3: Tl m 3 find the surface area of the sphere in terms of rr. \r _ 11-r:J ':> _ ~ r- 5 ~J~r i:: l.!.;1\ r'-? ::Z-lr'it L2) l.\'f -:. ~Z 'V "Z.. S,::.\b'\fN\ 'J - 3 ~ r - '0 )'\ 'D r~-::.g r -:.~ l t.. Example 6: The volume of a sphere is 7238 in 3. Find its surface area to the nearest number. \r - 1-\ ~ c-) 'J-~\\f =-1~'bo t.t1\r " :: ~\71~ 'l\ r? =- 5-4 ~~,0 f? -:..\/~7,qLl5ll'l r ~ \\,9qC1.~1.. rt;j\1- Example 7: C is the center of the sphere. Plane intersects the sphere in circle R. sa :::..'tf ( \"2.)2- whole ~ \<60 1,5573b8 SA ~ \<610 ~2 (0) Suppose CR = 5 and SR = 12. What is the length of a radius of the sphere? C, \3 (b) f the radius of the sphere is 41 and the radius of circle R is 40, find CR. "'l. '- -z... ~ '" + tto ~ We \ s 5 \2
More Practice 1. A 120 sector is cut out of a circular piece of tin with radius 6 in. and bent to form the lateral surface of ~ cone. What is the volume of the cone? h1", 2 4 ~ (.. 2 -t= ].":!ll _ "+1\ (L\"fi) (}0 ", - o.yc \(V\~~~ \~(\J.'l) /~(.. h~-;' ~l ~- ~ \1.0 ~,",'\( & 'h: llf'i -= \ \101\-{l..3 ~, 'c a.s e ci (: o M \ ~ kh. -------:--=----:-:--:-:::--;:;-~_: : :_::: '1'l\ ~ r ~.1-2. A circle with radius 12 feet is to be cut into congruent sectors and then the sec~ors will be made into cones._ Which will create the greatest volume capacity, 3 sectors of 120 or ~ secto~s of 90? CB ' & e, \ 12. u.1.-\,1~_\l.1., n.. o.m!-.hm.4t=-q(l'i1t')=o1r 1---.. v= -:a.. oj E9~'- '1'" (~'ii5') \,0 : ~~", \)'2.: \28 ; rz./~ a'i\("::~1'1j \ : '1 '{t"{\s. ~ 1'2.,/~~Y~8:q "'= ~ii. \ ~ ~ ~z. \ L.\c..o",e~:3fo'il"'~ a'fc' \er)~ n::~t'2.'\1\") r: 't \/::~; ll.'l\(8ii) \~.o 'it' " "'J:: \'\1.\,~q,,+ 'if - 8 11. 0 ~\-. \ +.::,?> '5' -. -.' \ ~~~ e s :' \ 2. ~ l1' ~., - \ ~ = ~~ 3. A right rectangular container is 5 cm. wide and 1L cm. long ana contams water to a depth of 7ern. A stone is placed in the water and the water rises 1.7cm. Find the volume of the stone. " 3 ~------~ 31 /1 "L_~ = (12)(5)(,)= '+20,-"" ~ / )., oc ore ~r\e = \OJ. em _ /~ \~A 5 \/o..fr'tf.f -:. (\' )(5)(8'."'):5:J.J.c.VV'~ \'2. J. A cone-shaped tank with base diameter 10ft and altitude 8 ft is being filled 'W/1ithwater at the rate of 18 fe per minute. How long will it take to fill the tank? ~,,:: as 11"~l'i) " t}.oo.,,'ff ~ a09.1\. '+.c.~3 ~ ~O'\;L\ ~ 1\. 61\ ~\".] 5. A can of tennis balls contains three balls. Find the ratio of the volume of the ban to the volume of the ' r~ ra.dhjs of c.'f. 'Vc~\,- ".r b:}= "" \ r ~V\_ ::' ~ ";.. =.s, (,f 2 ~ r:: V' o.ai us o~ b~\\ \o~\, = ~ 1f \,,3 \:xl\\s 1.\ 'Wr 2- ltj he\3hto~c~'=lor c'og.\\-s= 4-1rr 3 three balls.. \ - r:r 'l.( ", l' 3 'T'! ffi 6. A spherical tank whose radius to the outer surface is 15 ft is made of steel.5 in thick. How many cubic feet of steel are used in the construction of this tank? ' ',S~hefe. : ~ 't\ (\5)~= L\ 50011 (r:'\5),!' sf'-'et"-e. :: ~ 'it (1,\.q5g~) ~ L\L\lo~. ~DL\01 '\f (r: \L\o.q s&(3) 5 }v\ x,;...\~';"'-\-';""-- ':.0&\ \ Colo, \~~v\ \6~-\- - oq.(~~"\ s: \,-\.qs~~+-\-
7. A right rectangular container is 6 cm. wide and 15 ern. long and contains water to a depth of 5 em. A stone that has a volume of 6 cm'' is placed in the water. By how much does the fater level rise? (Round g '''',,= fa (\5')(5) - 0-5.0 0. '5 ~ L\-56 : go "f\. r]. - Lt ~ 0 e, tv) 5.0 fa = -R 0 b ~1O., C YY\ 8. A right cylindrical glass 8 em. in diameter contains water to depth of 3 em. Wh[ at volume of water must be added to raise the water level to 7 em.? ~ your answer to the nearest tenth.) _ (' D 5 01,, tt~~- b \5)"11 "'= \b\\(~)=4-~1t 'i -e \,10 'U (1.):' \\a 1\ U",,= ~o 'it'(8) 3 jo4-.~ [ ] if :\;}.9. ~15 m\~ -: do ~&' 1t if '1 9. Water runs into a right cylindrical tank at a rate of7 fe per minute. How long will it take to fill a tank 12 ft. in diameter that is 8 ft. high? (Round your answer to the nearest minute.) Xli 04..Rt/3 10. A cone-shaped tank with base radius 8 ft. and height 12 ft. is being filled with water at the rate of5 ftj per minute. How long will it take to fill the tank? & \J 13 5 \~ = b41r (\~ go4-. JS' -e,\ \loo.g mi\'\ \ - as lo \\ ~ ~ 0 L\. ;,)51/ 3 S 11. Find the volume and surface aresof a hemisphere of radius 10. ) '-a.+ \ ~ z.. '\ llirl.1..('\ ~ '\,:[000 3 \ 5 1\ e, 2. \.'\-'f' \0 -- 11" \0 bot\<> ~."" Q { \j = 2 "3 ' "10 ) = _,~ 'l'u. 1 = ;l.oo'lr -,,00 -rr of he ml P ""'10, 'S A- -=.3 10 'rr " 1 1 1. l6j2 n 3 2. 3 cones (y = 128J2 n fr') 3. vol of stone = 102 cnr' 4. Volume of cone = 662/3 n fe or 209.44 fr' Time to fill = 209.44 / 18 = 11.64 minutes or about 12 minutes 5. radius of ball = r radius of can is r and height of can is 6r V of can = 6m 3 V of one ball = 4/3 rrr3 so vol of 3 balls is 4nr 3 Therefore the ratio is 3 to 2 6. volume = 117.48 cubic feet 7. 0.1 em 8.24"u lo4-'\\ 9.129.25 minutes _ 10. 51.2n minutes.or about 160.8 minutes 11. V ~ 2000/3 n u' ; ja ~ 300 n u'
11-7 Study Guide - Areas and Volumes of Similar Solids Solids that have the same shape but different in size are said to be similar. by comparing the ratios of corresponding linear measurements. You can tell rf two solids are similar Determine if the two solids are similar. f so, give the similarity ratio. 1. ~ 2. Atd~ 16 in.! 10in. ltt. ': ~=~ ~ 5 :; r4==?l. ~5m. 8. 3 m. n. ~" 4in. 14 in.. d "1 Fill i th bj bel The following two cylm ers are sum ar. 1 m eta e ow. radius Circum- height Base area Latera] Volume ference Area Big i v. 8," \.. 12 LA.,\(" 1'\ u'2. 't G:,'f u2- let a 'fr U 3 Little.21.).. ft,'i\u Cfu.. Q1fu "'l 541t'u' 8\ 'it U 3 """',..,.26 em '.... ::: =::::-:.~.. 24 em 12 9 Linear ratios (similarity ~.3 ratio) Area ratios (!:i..,"2. _ ~ \.3') - q Volume ratios (.!:L\.3 _ ~.3) - a.! Given that these two cones are similar. 4) Find the similarity ratio. 5/ ~ 5) Find the ratio of their diameters. 5/ 3 6) Whatis the ratio of their base areas? (5~)'2.:::. ~ 7) What is the ratio of their volumes? (.. )3 =- ~ '~ ;lj 8) f the lateral area ofthe little cone is 60 in2, find the lateral area of the big cone. q'k..::: \500 ~ ~::. ~ \...it-\-\e. foo q 'lc:. \500 q 9) f the volume of the big cone is 600 em', find the volume of the little cone. ~12r. ~ ~ 1~5 1~5)C=.:t' «,;,(;10) L;'H-\e.. X ~..., ~ 5 ~ = 1(,.,'2.0 0 10) The ratio of the slant height of two pyramids is 2 to 5 and the surface area of the larger pyramid is.105 cm2. Find the surface area of the smaller pyramid. J. 5 'X,; 4- ;l. 0 \ineo.v- 2/5'.:L = ~ L.\ ~S \ 05 X ::: \ (0. R C lh 2- a. v e a. Y:lS 11) Two similar prisms have surface areas in a ratio of 9 to 16. f the volume of the smaller prism is 67.5 in', find the volume of the larger prism..;!1. to -, 5 1.3 -- -, 3 \iheo.-r Y'} ft.y. - X 'X ::: HoO i\"\ Q r eo, '\1(10.1.:1 x > 0 i (Co'1.5 ') '10\ ~ V /blf ~lx:::if3;to