PTR 10 Spe nd mesuement PTR ONTNTS 10 Pytgos teoem 10 Pytgos teoem in tee dimensions 10 Peimete nd e 10 Totl sufce e (TS) 10 Volume 10F pcity 10G Simil figues 10 Simil tingles 10I Symmety IGIT O doc-9608 10 Quick Questions 10 Pytgos teoem In igt-ngled tingle te longest side (wic is opposite to te igt-ngle) is clled te ypotenuse. Fo ny igt-ngled tingle wit side lengts, nd ypotenuse c it cn e poved tt Tis is clled Pytgos teoem. + = c ypotenuse c WORK XMP 1 Find te vlue of te ponumels to 1 deciml plce. TINK 1 In Δ only is known; teefoe sould e found fist. WRIT Use Δ to find using Pytgos teoem. Δ: = 5 + 17 3 Wit te dded infomtion, find y. Rememe tt = 914. = 65 + 89 = 914 = 914 30. Δ: ( 914 ) = y + 1 914 = y + 144 y = 914 144 y = 770 = 770 = 7.748 873 85 4 Round to te equied nume of deciml plces. y 7.7 5 17 y 1 PTR 10 Spe nd mesuement 379
IGIT OS doc-9609 SkillST 10.1 Pytgos teoem doc-9610 Pytgos teoem ecise 10 Pytgos teoem 1 Fo ec of te following tingles, find te lengt of te unknown side coect to 1 deciml plce. 4 1 10 9 c 5 d 4. 6.3 c 19 d e e f f 16. 14.8 6.7 Wt is te lengt of digonl of: sque wit side lengts of 6 cm (coect to 1 deciml plce) ectngle wit dimensions of 0.8 m nd 37 cm? (Give you nswe in metes coect to deciml plces.) 3 W1 Find te vlue of te ponumel to 1 deciml plce in ec of te following spes. 3.8 1.8 m.5 m 16 1. m 1.4 m 4 long-distnce unne wnts to complete 30-km un. Se s completed 18 km nd is t te mked point. If se took te sot cut coss te pk, would se ec e gol o sould se go long te steets? Justify you nswe wit mtemticl evidence. ome My St y Pk cossing 4.5 km km 5 km 5 km Runne mpton St 5 Two of te legs of titlon e set in tingul fomt. Te swim Te swim leg s te competitos eding out due not to uoy 1, tuning due est to te second uoy nd ten eding stigt ck to te stting position. w digm of tis sitution. lculte te totl lengt of te swim if te fist two legs e 300 m nd 400 m espectively. Te cycle Te tletes leve te tnsition e nd ide in soutely diection fo 1 km. Tey ten tun due west nd ide fo 16 km. Te lst leg s te cyclists eding stigt ck to te tnsition e. c ompetito 4 s cycle develops punctue t te stt of te lst leg. ow f does competito 4 ve to wlk to get ck to te tnsition e? 380 Mts Quest 11 Stndd Genel Mtemtics
10 Pytgos teoem in tee dimensions Tee-dimensionl ojects cn ve igt-ngled tingles witin tem tt cn e e-dwn in dimensions, so tt Pytgos teoem cn e used to find missing mesuements. WORK XMP Find te lengt lelled. Round you nswe to 1 deciml plce. TINK 1 Identify igt-ngled tingle tt contins nd edw it in two dimensions. WRIT/RW 1 1 8 17 F Wite Pytgos fomul. 1 = 8 + 3 Solve fo. = 1 8 4 Given tt is te side lengt of tingle ten its vlue is positive. 8 = 144 64 = 80 = ± 80 = 8.9447. 5 Wite te nswe, coect to 1 deciml plce. Te eigt lelled is 8.9 units. WORK XMP 3 is sque pymid. Find, te eigt of te pymid. TINK WRIT/RW 45 1 () Te unknown lengt () cn e found fom tingle. () In tis tingle, te lengt of is missing s well so it s to e found fist. Te lengt of is lf te lengt of. (c) In tun, cn e found fom tingle. Redw tis tingle in dimensions nd include ll y mesuements. 0 0 0 Use Pytgos teoem to find. eve te nswe in sud fom. c = + y = 0 + 0 = 400 + 400 = 800 y = 800 = 400 = 0 PTR 10 Spe nd mesuement 381
3 Find te lengt of y lving te lengt of. (Tee is no need to evlute 0, s we will ve to sque tis vlue in te steps tt follow.) = 1 1 = 0 = 10 4 Redw te tingle tt contins te unknown lengt in dimensions (tingle ). Wite te mesuements, including te lengt of found in te pevious step. 45 10 5 Use Pytgos teoem to find. (Round you nswe to 1 deciml plce.) c = + c = 45, =, = 10 45 = + (10 ) 05 = + 00 = 05 00 = 185 = 185 4.7 IGIT O doc-9610 Pytgos teoem ecise 10 Pytgos teoem in tee dimensions 1 W In ec of te following, find te lengt of te line lelled. Round you nswes to 1 deciml plce. F F G 10 4 15 0 G 8 15 c d 6.6 8. 16 1 W3 In ec of te following, find te lengt of te line lelled. Round you nswes to 1 deciml plce. G c d 8 0 F 5 9 1 16 10 15 6 3 Te digm t igt sows ectngul metl pencil cse. Find: te lengt of te longest pen tt cn e plced flt on te ottom of te pencil cse te lengt of te longest pen tt cn e plced in tis pencil cse. 1 0 cm 15 cm 10 cm 38 Mts Quest 11 Stndd Genel Mtemtics
10 Peimete nd e Te tle elow sows te fomuls fo finding te e nd peimete of some common spes. Spe e Peimete Sque = P = 4 Rectngle = W P = ( + W) W Pllelogm = wee te eigt mesuement must e t igt ngle to te se mesuement. P = sum of ll sides Tpezium = 1 ( + ) wee te eigt mesuement must e t igt ngle to te se mesuement. P = sum of ll sides Romus y = 1 y P = sum of ll sides Tingle = 1 wee te eigt mesuement must e t igt ngle to te se mesuement. P = sum of ll sides Tingle c = s( s )( s )( s c) wee s = 1 ( + + c) (Use wen eigt mesuement is unknown.) P = + + c icle = π = π d o = π d PTR 10 Spe nd mesuement 383
WORK XMP 4 lculte te e nd te peimete of tis spe. 7 m 8 m TINK WRIT 10 m 1 Wite te ppopite fomul fo te e. = s( s )( s )( s c) wee s = 1 ( + + c) Identify te vlues of te ponumels. = 7, = 8, c = 10 3 To find s, sustitute, nd c vlues into te fomul nd simplify. 4 Sustitute te vlues of,, c, nd s into te fomul fo te e. 5 vlute. (Round te nswe to 1 deciml plce nd include te units.) 1 Find te peimete y dding ll te side mesuements. s = 1 (7 + 8 + 10) = 1 5 = 1.5 = 1.5(1.5 7)(1.5 8)(1.5 10) = 1.5 5.5 4.5.5 = 773.4375 = 7.810 744 33 = 7.8 m P = 7 + 8 + 10 = 5 Wite te nswe, including te ppopite units. P = 5 m omposite figues Te tem composite mens mde up of distinct pts. omposite figues in geomety e figues compising nume of distinct spes. epending upon te composite figue, to find te ovell e o peimete you my need to dd tese individul spes o sutct one fom note. e of n nnulus icles e sid to e concentic if tey ve te sme cente point. Te e etween te two concentic cicles is efeed to s n nnulus. e of nnulus = e of lge cicle e of smlle cicle = π R π = π (R ) wee R = dius of te lge cicle = dius of te smll cicle. nnulus 1 R WORK XMP 5 In one full evolution, te 6-cm-long minute nd of clock would sweep out lge cicle tn te 3-cm-long ou nd. Wt is te diffeence in te e tey cove to te neest sque centimete? TINK WRIT TUTORI eles-1384 Woked emple 5 1 Te e equied is te nnulus. Wite te ppopite fomul. Identify te vlue of R (dius of te lge cicle) nd te vlue of (dius of te smlle cicle). = π (R ) R = 6, = 3 6 cm 3 cm 3 Sustitute te vlues of te ponumels into te fomul. = π (6 3 ) 384 Mts Quest 11 Stndd Genel Mtemtics
4 vlute. = 84.83 5 Wite n nswe sentence wit te vlue ounded to te neest sque centimete. Te diffeence in e coveed y te two nds is ppoimtely 85 cm. e of secto nd c lengt secto of cicle cn e tougt of s wedgesped slice of pie. Te e of te secto cn e detemined y finding te fction of te wole cicle it epesents. θ e of secto = π, 360 wee θ is te ngle of te secto nd is te dius of te cicle. O θ Secto of cicle WORK XMP 6 10-cm-long minute nd moving fom te nume 1 to te nume 4 position sweeps out secto. Wt is te e of tis secto? TINK WRIT 1 Wite te fomul fo te e of te secto. e of secto = Identify te vlue of te dius. = 10 θ π 360 1 1 10 cm 3 4 3 lculte te ngle of te secto: Te ngle etween consecutive numes on clock = 360 1 = 30. Fom 1 to 4 tee e fou intevls etween te numes. So to find te ngle of secto, multiply 30 y 4. 4 Sustitute te vlues of nd θ into te fomul nd evlute. 5 Wite n nswe sentence wit te nume ounded ppopitely nd units given. θ = 30 4 = 10 e of secto = 10 360 π 10 = 104.719 755 1 Te minute nd s it ottes toug n ngle of 10 sweeps toug n e of 104.7 cm. Te cuved edge of secto is clled n c. θ c lengt = π d 360 o θ c lengt = π 360 wee θ is te ngle of secto, d is te dimete of cicle is te dius of cicle. θ c PTR 10 Spe nd mesuement 385
WORK XMP 7 Wt distnce did te tip end of te 10-cm minute nd tvel wen it moved fom pointing to te nume 1 to pointing to te nume 4? TINK 1 Since te dius is known, wite te fomul fo c lengt involving te dius. WRIT Stte te vlue of. = 10 c lengt = θ π 360 1 1 10 cm 3 4 3 Find te size of te ngle of secto. () Te ngle etween ec nume on clock = 30. () Tee e 4 intevls etween te numes; teefoe te ngle of te secto cn e found y multiplying 30 y 4. 4 Sustitute vlues of nd θ into te fomul nd evlute. 5 Wite n nswe sentence ounding te vlue ppopitely nd witing in te units. θ = 30 4 = 10 10 c lengt = π 360 10 = 0.943 951 0 Te tip of te minute nd tvelled 0.9 cm. IGIT O doc-9611 SkillST 10. onvesion of units lengt IGIT O doc-961 SkillST 10.3 e nd peimete of composite spes ecise 10 Peimete nd e 1 W4 Find: i te e nd ii te peimete of te following spes, to deciml plces. d 0 mm 4 cm 18 cm 14 mm 140 cm 16 cm e 1 cm 5.5 m m.5 m 1.5 m c 1 cm 5 cm 9 cm lculte i te e nd ii te peimete of te following spes. Give you nswes to deciml plces. 5 cm 10 cm 15 cm 13 cm 30 cm 6 cm 3 M mine te digm t igt. 3 cm Te cicles cove n e of ppoimtely: 40 cm 01 cm 804 cm 805 cm 603 cm Te sded e is ppoimtely: 19 cm 41 cm 6 cm 83 cm 0 cm c metl mnufctue is le to cut only fou discs fom evey seet of metl. Wt pecentge of metl is wsted? 80% 1% % 41% 61% 10 m c 386 Mts Quest 11 Stndd Genel Mtemtics
4 gud dog inside used c slesyd is tied to cone post of te fence suounding te yd. Te fence sides meet t igt ngle nd te dog is on ope 1. m long. i w digm of tis sitution sowing te e ccessile to te dog. ii To 1 deciml plce, ow muc e does te dog ve in wic to eecise? One nigt te dog is moved to te outside cone of smll ectngul uilding mesuing m 6 m. Te dog s ope s een lengtened to 3 m. i w digm of tis sitution sowing te e ccessile to te dog. ii lculte te e ville fo te dog fo eecise. Give you nswe to te neest sque mete. c If te owne ties te dog ck to te fence post nd e wnts te dog to ve s muc oom to eecise in s it did wen it ws tied to te uilding, wt lengt of ope (to te neest centimete) does e need to pucse? Geomety is used etensively in design. Questions 5 10 eploe te use of geometicl spes in te design of county flgs, compny logos, nd some commonly used signs. 5 Te flg of Jpn is ed cicle on wite ckgound. lculte (to 1 deciml plce) te dius of te cicle, if te e of te cicle must e 0% of te totl flg e. 6 Te ommonwelt nk logo is mde up of yellow sque wit lck tpezium ovelid t one cone. lculte, to te neest wole nume, wt pecentge of te ovell design is lck. 7 W5 Te collectle plte sown elow is cm in dimete nd s golden 0.5-cm-wide ing. cm 1 cm 0.5 cm 3.0 cm.5 cm 3 m 1 cm m Find (to 1 deciml plce) te e of te golden ing if its oute edge is 1 cm fom te edge of te plte. 8 W6 fmily-size pizz is cut into 8 equl slices. If te dimete of te pizz is 33 cm, find (to te neest sque centimete) te e of te top pt of ec slice. 9 W7 Using te mesuements given in te digm t igt, find te lengt of te ike cin. 10 ly-model veicles d single windsceen-wipe lde to emove wte fom te windsceen. (Te us t igt s two single ldes of tis type.) Using te dimensions given in te digm elow igt: wt e (to te neest wole nume) did te lde cove? wt pecentge (to 1 deciml plce) of te windsceen ws cleed? c wt distnce (to te neest wole nume) does te tip of te lde tvel in one full sweep? 4 cm 10 cm 6 cm 150 10 cm 160 60 cm 140 45 cm PTR 10 Spe nd mesuement 387
10 Totl sufce e (TS) To find te totl sufce e (TS) of 3-dimensionl oject, we find te e of ll te sufces, nd dd tem togete. lcultion of te totl sufce e of tese uildings is comple tsk. Fo some common ojects, we ve fomul. Oject Net TS ue TS = 6 1 3 4 5 6 Rectngul pism l TS = (w + lw + l) l w w l w ylinde π TS = e of cicles + cuved sufce = π + π = π ( + ) Spee Not sown TS = 4π 388 Mts Quest 11 Stndd Genel Mtemtics
Oject Net TS one S S = Slnt eigt π TS = e of se (cicle) + e of cuved sufce = π + π S = π ( + S) Sque-sed pymid TS = e of sque + e of 4 tingles = + 4 1 = + WORK XMP 8 Find te totl sufce e of te oject sown t igt. TINK WRIT 17 cm 1 Identify te spe. (Spe: ectngul pism) Wite te fomul fo te TS of ectngul pism. TS = (w + lw + l) 19 cm 9 cm 3 llocte vlue to te ponumels. w = 9, = 17, l = 19 4 Sustitute te vlues of te ponumels into te fomul. TS = (9 17 + 19 9 + 19 17) 5 vlute (ckets fist, ten multiply y ). = (153 + 171 + 33) = 647 = 194 6 Wite te nswe, including units. TS = 194 cm In some situtions you my know te totl sufce e of n oject ut e missing dimension. In tis cse, you cn use te totl sufce fomul to find te unknown dimension. WORK XMP 9 tennis ll s sufce e of 154 cm. Will it fit toug cicul ole wit dimete of 6 cm? TINK WRIT 1 Wite te fomul fo te TS of spee. TS = 4π llocte te ponumels vlue. TS = 154 =? 3 Sustitute known vlues into te fomul. 154 = 4π 4 vlute. Solving 154 = 4 π fo gives = 3.5007 o = 3.5007. 5 Given tt is te dius of tennis ll, its vlue is positive. Since > 0, = 3.5007. PTR 10 Spe nd mesuement 389
6 Te question equies te dimete of te tennis ll so multiply te dius y. 7 ompe te dimete of te ole wit te dimete of te ll nd wite n nswe sentence. d = = 3.5007 7.0 Te tennis ll will not fit toug te cicul ole ecuse its dimete is ppoimtely 7.0 cm, wile te ole s dimete is 6 cm. Mny uildings e composite figues mde up of pisms nd pymids. WORK XMP 10 Te digm sows te poposed spe fo new contine fo tkewy inese food. Find te TS of te contine. 10 cm TUTORI eles-1385 Woked emple 10 TINK WRIT 10 cm 1 Identify te distinct spes tt mke up te totl oject: tese e sque-sed pymid nd cue. Te se of te pymid nd one fce of te cue e not on te sufce nd teefoe tei e sould not e included. TS = 4 tingles + 5 sques. lculte te e of te tingles. = 4 1 = = 10, = 10 = 10 10 = 00 390 Mts Quest 11 Stndd Genel Mtemtics
3 lculte te e of te sques. = 5 4 dd te individul TS togete to find te TS of te wole oject. = 10 = 5 10 = 500 TS = 00 + 500 = 700 cm ecise 10 Totl sufce e (TS) 1 W8 Find te totl sufce e of te following ojects. Round you nswe to 1 deciml plce. 0 cm c 1. cm 10 cm 7.5 cm 3.5 cm 6. m d 10.5 m e 3 cm f 14 mm 8.4 m 9.0 cm 16 cm 4 mm Mtc te fomuls elow wit te oject. l w c S (se not included) Open-ended cone d e f g i S se included l i TS = π + π = π ( + ) iii TS = 4( 1 ) ii TS = π S iv TS = π ( + S) v TS = 4π vi TS = (w + lw + l) vii TS = ( 1 ) 4π i TS = = π + (3 l) viii TS = 1 [π ( + )] + = π ( + ) + PTR 10 Spe nd mesuement 391
3 Fo te following digms, find te vlue of S fo te fomul: TS (cone) = π + π S = π ( + S) 10.4 cm 6 cm 1 cm 4 Fo ec of te following ojects find te vlue of te ponumel, ounding you nswe to 1 deciml plce wen equied. 4 m w =? 6 m TS = 148 m 3 m =? TS = 188.5 m c 9 m 7 m d = 6 cm s =? TS = 100 cm d =? TS of 1 spee = 55.4 cm 5 W9 cylindicl cok is 6 cm ig nd s TS of 33.38 cm. ould it e used to close ottle wose neck is 3 cm in dimete? Justify you nswe. 60 cm 6 M 60-cm-ig cone s se dius of 3 cm. Te TS could e clculted y using te fomul: π 3(3 + 60) π 60(3 + 60) π 3(3 + 68) π 68(3 + 68) π 60(3 + 68) 3 cm 7 M speicl cndle of TS = 01 cm is to e gift-oed. Te dimensions in centimetes of sevel diffeently sped oes e given elow (lengt eigt widt). Te o wic will est fit te cndle wit te lest mount of wsted spce is: 6 6 6 4 4 6 8 4 4 8 4 8 8 8 8 M Te e coveed y te olling pin sown in one complete tun is: π 5(5 + 30) π.5(.5 + 30) π.5(.5 + 30) π.5 30 π 5 30 9 Wt e of cdod needs to e pucsed to constuct o 1 m 1 m 1 m to contin television set? llow n et 5% of te totl sufce e to cte fo ovelps. 5 cm 30 cm IGIT OS doc-9614 WokST 10.1 Investigtion doc-9613 one eds 10 Wt is te nme of te spe ( type of pism) of tis cocolte pckge? In you mind, unfold te 3-dimensionl spe to its -dimensionl net. w tis net, mking no llownce fo ovelp. c Using te dimensions given, clculte te ppoimte TS of te cocolte pckge. (Round ll vlues to 1 deciml plce.) 11 W10 etin medicines come in cpsules, s sown t igt. Find te e of plstic (in sque millimetes) needed to poduce one suc cpsule. (Ignoe ovelp). 1 commecil ed in s dimensions s sown in te digm t igt. Find te TS of te in. 90 cm 60 cm 19.5 cm 3.5 cm 5 mm 10 mm 39 Mts Quest 11 Stndd Genel Mtemtics
10 Volume Te volume of n oject is te mount of spce tt te oject occupies. Volume of pism = coss-sectionl e eigt of te pism V = Te eigt is te dimension pependicul to te coss-sectionl e. Spe oss-sectionl spe Volume ylinde e = π V = e of cicle eigt = π Tingul pism e = 1 V = e of tingle eigt = 1 Note: owecse epesents te eigt of te tingle. Rectngul pism W W e = W V = e of ectngle eigt = W ue e = V = e of sque eigt = = = 3 (since in cue, = ) WORK XMP 11 Find te volume of te spe sown coect to 1 deciml plce. TINK WRIT 1 Identify te spe. Tingul pism.3 m.6 m 3. m Wite te ppopite fomul fo te volume. 3 llocte vlues to te ponumels keeping in mind tt nd e te se nd eigt of tingul coss-section o se of te pism, wile is te eigt of te pism. 4 Sustitute nd evlute, ounding te nswe to 1 deciml plce. Include te units. V = 1 =.6, =.3, = 3. V = 1.6.3 3. = 9.568 9.6 m 3 PTR 10 Spe nd mesuement 393
Odd-sped pisms Te oject t igt is pism, ecuse oizontl cuts sow unifom coss-section. To clculte te volume of tis pism you would need to e given te vlue fo te coss-sectionl e nd te eigt. eigt oss-section WORK XMP 1 Find te volume of te spe sown t igt. TINK 1 Wite te genel fomul fo te volume of pism. WRIT V = eigt llocte te ponumels vlue. = 3 m, = 8. m 3 Sustitute te vlues of te ponumels into te fomul nd evlute. V = 3 8. = 6.4 8. m e = 3 m 4 Wite te nswe including units. V = 6.4 m 3 Pymids pymid s flt se t one end nd point t te ote. one Sque pymid Rectngul pymid Tingul pymid pymid occupies ectly one tid of te spce tken y pism of te sme se nd eigt. Volume of pymid = 1 3 e of se eigt of oject = 1 3 Te following tle sows te fomuls fo te volume of some common pymids. Spe Flt end (se) spe Volume one V = 1 3 e of cicle eigt V = 1 3π Sque pymid V = 1 3 e of sque eigt V = 1 3 394 Mts Quest 11 Stndd Genel Mtemtics
Spe Flt end (se) spe Volume Rectngul pymid W V = 1 3 e of ectngle eigt = 1 3 W W Tingul pymid V = 1 3 e of tingle eigt 1 1 V = 3 Note: owecse epesents te eigt of te tingle. Spees Te volume of spee is given y te following fomul: Volume of spee = 4 3π 3 wee is te dius of te spee. emispee is lf of spee. Its volume, teefoe, is lf of te volume of spee. Volume of emispee = 1 (volume of spee) omposite solids = 1 4 3π 3 = 3π 3 Te volumes of te individul ojects need to e found efoe tey e dded o sutcted to find te totl volume. WORK XMP 13 Find te volume (in m 3 ) of te toy sown, coect to 1 deciml plce. TINK WRIT 1 Identify te components of te spe. Totl volume = volume of cone + volume of emispee etemine te volume of te emispee. () Wite te fomul. () llocte vlues to te ponumels. Note tt some units e in cm, otes in m. s te nswe equies cuic metes, cnge centimetes to metes. (c) Sustitute te vlue of into te fomul nd evlute. Volume of emispee: V = 3π 3 = d = 80 = 40 cm = 0.4 m V = 3π (0.4) 3 = 0.134 041 3 m 3 TUTORI eles-1386 Woked emple 13 1.4 m 80 cm PTR 10 Spe nd mesuement 395
3 Find te volume of te cone. () Wite te fomul. () llocte vlues to te ponumels. (eigt = 1.4 dius of cicle) Volume of cone: V = 1 3π = 0.4 m, 4 Sustitute te vlues of nd into te fomul nd evlute. V = 1 3π (0.4) 1 = 0.167 551 6 m 3 5 dd te individul volumes togete to find te totl volume of te given spe. 6 Round you nswe to 1 deciml plce nd include units. = 1.4 = 1.4 0.4 = 1 m Totl volume = 0.134 041 3 + 0.167 551 6 = 0.301 59 9 V = 0.3 m 3 ecise 10 Volume 1 Mtc te volume fomul wit te ppopite 3-dimensionl oject. c cm d e f i V = 1 ii V = 1 3π iii V = ( + ) 1 IGIT O doc-9615 SkillST 10.4 Volume iv V = π v V = 1 3 vi V = W 11 Find te volume of ec of te following spes coect to 1 deciml plce. 8 cm 6 cm 10. cm 7.4 cm c 4 cm 10 cm d.4 cm e 1.6 cm 18.5 cm 1 cm 396 Mts Quest 11 Stndd Genel Mtemtics
3 W1 Find te volume of ec of te following spes. Sded e = 3 cm 8 cm Sded e = 4 cm 3.5 cm c d 10.5 mm Sded e = 116 mm 0.5 m Sded e = 55 cm 4 lende is odeing concete se (in te spe of te tpezoidl 60 cm pism sown t igt) fo is fvouite gden sculptue. ow muc will e ve to py if concete costs $50 pe cuic mete nd te cost of lou is $45? 1. m 5 W13 Find te volume of ec of te following spes coect to 1 deciml plce. 60 cm c 10 cm 50 cm 50 cm IGIT O doc-9616 SkillST 10.5 onvesion of units volume 1.4 m m 1.8 m 113 cm 64 cm cm 5 cm 6 Te digm t igt sows 3 tennis lls pcked in cylindicl contine. Find: te volume of ec ll te volume of te cylinde c te volume of spce tt emins fee. 7 cocolte compny wnts to mke cocolte istms ll decotions contining 50 smll, cndy-coted cocoltes to ng on istms tees. If ec cndy-coted cocolte s volume of ppoimtely 0.8 cm 3, wt is te dimete of te istms ll equied to contin tem? 8 lge pple tkes up ppoimtely 51 cm 3 of spce. I ve 160 pples to pck into one of te following contines. Wit te im of ving te miniml mount of wsted spce, wic contine would e est fo tis pupose nd wy? 40 cm 38 cm 40 cm c 38 cm 15 cm 40 cm 7 cm IGIT O doc-9617 SkillST 10.6 Finding unknown lengts 60 cm 38 cm 70 cm 4 cm 34. cm 9 Te flowe vse sown elow is fom designe sop. lculte te totl volume of te vse, ounding ll clcultions to 1 deciml plce s you go. 18 cm 4 cm 1 cm IGIT OS doc-9618 Investigtion one volume doc-9619 one volume PTR 10 Spe nd mesuement 397
10F pcity Te cpcity of contine efes to te mount tt it cn old. Te cpcity o volume of contine is usully mesued in cuic units; oweve, wen te volume of liquid is eing discussed it cn e efeed to in tems of millilites, lites nd kilolites. Recll te following fcts: 1000 millilites (m) = 1 lite () 1000 lites = 1 kilolite (k) uic units e elted to te fluid cpcity units s follows: 1 cm 3 = 1 m 1000 cm 3 = 1 1 m 3 = 1000 = 1 k WORK XMP 14 onvet: 400 cm 3 to m 100 cm 3 to m nd to c k to m 3. TINK Since 1 cm 3 is equivlent to 1 m, ten 400 cm 3 is equivlent to 400 m. WRIT 400 cm 3 = 400 m 1 c 1 cm 3 will old 1 m of liquid. Teefoe, 100 cm 3 will old 100 m of liquid. To cnge m to, divide y 1000 (since tee e 1000 m in 1 ). c One k is equivlent to 1 m 3. Teefoe, k is equivlent to m 3. 100 cm 3 = 100 m = 1. c k = m 3 To find te cpcity of contine in lites, find its volume in cuic units fist nd ten convet. WORK XMP 15 Find te cpcity in m of ectngul contine mesuing 10 cm 1 cm 14 cm. TINK 1 Find te volume of te contine in cm 3. (Since te contine is ectngul, use te fomul fo te volume of ectngul pism.) WRIT V = l w = 10 1 14 = 1680 cm 3 nge cuic centimetes to millilites. 1680 cm 3 = 1680 m 3 Wite te nswe in wods. Te cpcity of te given contine is 1680 m. ecise 10F pcity IGIT O doc-960 pcity 1 W 14 onvet te following units s indicted. 750 cm 3 = m 800 cm 3 = c 500 cm 3 = m d 40 000 cm 3 = e 6 m 3 = cm 3 = m = f 1 m 3 = g 4. m 3 = k 7.5 m 3 = k = i 5. m = cm 3 j 6 = cm 3 k 0 = m = cm 3 l 5.3 K = m 3 398 Mts Quest 11 Stndd Genel Mtemtics
W 15 ow mny millilites will ectngul dink contine old if its dimensions e 11 cm 4 cm 15 cm? 3 emispeicl owl wit dimete of 30 cm will e used to old pe-mied fuit dink fo pty. If you wnt to fill it to te im, ow mny lites of pe-mied dink cn you pou in? 4 One lite of onge nd mngo juice is pcked in ectngul contine of eigt 17 mm nd widt 93 mm. Find te lengt of te contine to te neest mm. 5 plstic ottle contins 1.5 of soft dink. ow mny cylindicl cups 10 cm ig nd 7 cm in dimete cn e filled to cpcity fom tis ottle? Wt is te volume (to te neest m) of te dink tt emins in te ottle? 6 ectngul swimming pool mesues 4 m y 3.5 m y.5 m. Wt is te cpcity of te pool in k? If te pool is eing filled t te of 14 pe second, ow long will it tke fo it to e: i 70% full? ii filled to cpcity? 7 tu in te spe of tpezoidl pism s dimensions s sown. If te cpcity of te tu is 943.5, ow deep is it? 0.85 m 17 mm d l 30 cm. m 93 mm 10G Simil figues Imgine tt you wnted to enlge Δ y fcto of. You could plce it unde potocopie set t 00%. O you could dw it youself. Fist dw 6 cm. Mke equl to 90. w 8 cm. Join. ecking, you will find tt is 10 cm. 6 cm 1.5 m 3 cm 5 cm 4 cm 8 cm Te tingles ve identicl ngles, ut te sides in te second tingle e twice s lge. Figues like tese, wic ve ectly te sme spe e clled simil fi gues. We sy tt Δ is simil to Δ, nd wite Δ ~ Δ Δ s een enlged y scle fcto of. eck te e of ec figue. 3 Δ: e = 4 6 Δ : = 8 e = 6 = 4 Te e s een incesed y fcto of. In genel, if figue is enlged y fcto of, ten its e will incese y fcto of. Te following emple sows ow figue cn e enlged fom fied point. WORK XMP 16 nlge tis spe y scle fcto of. PTR 10 Spe nd mesuement 399
TINK 1 Select point inside te spe, sy, point O nd dw lines fom tis fied point to te spe s vetices (tt is points,, nd ). Mesue te lengts of O, O, O nd O. Note: Te mesuements e in cm. 3 Multiply ec of te mesuements y scle fcto of. 4 tend te lines to te new mesuements; tt is, continue line O until it mesues.1 cm, line O until it mesues 3.5 cm nd so on. 5 Join te ends of te etended lines to fom te simil figue. el te vetices ppopitely. WRIT/RW O O = 0.95, O = 1.05, O = 1.75, O = 1.75 O = 0.95 O = 1.75 = 1.9 = 3.50 O = 1.05 O = 1.75 =.1 = 3.50 ' ' O ' ' Te following emple sows ow to educe figue fom fied point. WORK XMP 17 Reduce te pentgon lelled y scle fcto of. TUTORI eles-1387 Woked emple 17 TINK 1 Select point outside te spe nd lel it O. w lines fom tis fied point O to te spe s vetices nd mesue tei lengts. Note: Te mesuements e in cm. WRIT/RW O O = 6.8, O = 7.6, O = 7.1, O = 5.8, O = 5.6 ivide tese mesuements y te scle fcto of. O = 6.8 O = 7.1 O = 5.6 = 3.4 = 3.55 =.8 O = 7.6 O = 5.8 = 3.8 =.9 3 octe te vetices of educed pentgon y mesuing new distnces (tt is O, O nd so on) fom point O long te ' eisting lines. ' ' 4 Join te vetices,,, O nd to fom educed figue. ' ' 400 Mts Quest 11 Stndd Genel Mtemtics
ecise 10G Simil figues 1 W16 nlge ec of te following spes y te given scle fcto. Scle fcto = 3 Scle fcto = 1.6 O O W17 Reduce te following spe y te given scle fcto. Scle fcto = Questions 3 to 7 will elp you to investigte te e nd volume tios of simil figues. 3 Wic of te following figues e simil? ll ectngles ll sques c ll pentgons d ll cicles e ll equiltel tingles f ll isosceles tingles g ll igt-ngled tingles ll isosceles igt-ngled tingles Mtemticlly it cn e sown tt if te side lengts of two simil figues e in te tio : ten te es of te simil figues e in te tio :. Sques e lwys simil figues. Fo te two sques sown: if te side lengt tio = :6 ten te e tio = :6 = 4:36 = 1:9. Teefoe, if te e of te smlle sque is 4 cm, ten te e of te lge sque is: lge sque e = scle fcto (tio) smll sque e lge sque e = 9 4 1 lge sque e = 36 cm 6 4 opy te simil ectngles t igt into you wokook. opy nd complete te following sttements nd clcultions. Smlle:ge Smlle:ge Side lengt tio :? o? :10 e tio :?? :10 = 4:? =? :100 =? :5 It does not mtte wic pi of dimensions you coose; te e tio will finlly wok out to e te sme. If te e of te lge ectngle is 50 cm, use te e tio to clculte te e of te smlle ectngle: e of smll ectngle = scle fcto e of lge ectngle e of smll ectngle = 4 50 5 e of smll ectngle =? cm eck: e of ectngle = W = 4, W = = 4 =? cm F 1 4 10 5 PTR 10 Spe nd mesuement 401
5 Sow tt Δ ΔF. Wt is te tio of te two tingles lengts in simplest fom? c Wt is te tio of tei es? d If te e of Δ is 6 cm, use te e tio to sow tt te e of ΔF is 13.5 cm. 6 w te following cylindes into you wokook. 4 3 5 6 4.5 7.5 F 6 cm = 5 cm 10 cm = 3 cm IGIT OS doc-961 WokST 10. doc-96 Investigtion Mking n eoplne INTRTIVITY int-0811 Simil tingles opy nd complete te following sttements nd clcultions. Using te eigt dimensions: Using te dius dimensions: Side lengt tio? :10 o 3:? Volume tio? :10 3 o 3 3 :? (vlute)? :1000 o 7:? (ncel)? :15 If te volume of te lge cylinde is ppoimtely 785.4 cm 3, use te volume tio to clculte te volume of te smlle cylinde to 1 deciml plce: Volume of smll cylinde = scle fcto Volume of lge cylinde? Volume of lge cylinde = 7 15 Volume of smll cylinde =? cm 3 c eck tis vlue y using te fomul: volume = e of coss-section eigt. 7 Wt is te volume tio (in simplest fom) etween te cones sown t igt? If te volume of te lge cone (to 1 deciml plce) is 56.5 cm 3, use te volume tio to clculte te volume of te smlle cone to 1 deciml plce. c eck you nswe using te fomul fo te volume of cone. 10 Simil tingles Tee e tee foml test ules tt cn e pplied to tingles to see if tey e simil. 9 6 4.5 80 3 40 60 4 F cm G 80 4 cm 3 cm 6 cm 8 40 60 I Test 1 () Tingles e sid to e simil if ll te coesponding ngles e equl. Tt is, te tee ngles in one of te tingles e equl to te tee ngles in te ote tingle. ΔF ΔGI, ecuse = G = F = I Note: If two pis of coesponding ngles e equl, ten te 3d pi is lso equl. 40 Mts Quest 11 Stndd Genel Mtemtics
Test (SSS) Tingles e sid to e simil if te tios etween te coesponding side lengts e equl. Δ ΔF, ecuse te tio of coesponding side lengts is : 9 = =, 6 8 = =, nd = = 4.5 F 3 Test 3 (SS) Tingles e sid to e simil if two of tei coesponding sides e in te sme tio, nd te ngle etween tese two sides (te included ngle) is te sme in ot tingles. Δ ΔF, F 4 18 = = 3 nd = = 3; = 8 6 Sometimes it is d to decide wete te two tingles e simil o not, ecuse tey e not oientted te sme wy. In suc cses it is elpful to e-dw te tingles so tt te sides nd ngles tt we tink migt e coesponding e in te sme ode. 6 F 60 8 4 18 60 4 Included ngle F WORK XMP 18 ompe ec of te tingles t igt wit Δ nd stte wete tey e simil, o if tee is not enoug infomtion given fo decision to e mde. Justify you nswes. Δ F ΔGI c Δ JK d Δ MNO F TINK 40 0 N 6.6 6 110 40 30 30 10 J G 0 7.5 6.6 10 110 110 40 110 K 4.5 I WRIT/RW 110 O M 1 Re-dw ΔF so tt its ngles coespond to tose of Δ. 110 6.6 6 30 10 40 30 110 40 0 F ompe te coesponding ngles nd wite wt you oseve. 3 Stte wete te tingles e simil nd specify te test upon wic you conclusion is sed. = = = F ll coesponding ngles e equl. Δ ΔF (). 1 Re-dw Δ nd ΔGI so tt ngles/sides coespond. 30 6.6 110 10 6 40 G 110 7.5 4.5 I 40 ompe te side mesuements. 6 = I 4.5 = 1.333 nd 10 = GI 7.5 = 1.333 PTR 10 Spe nd mesuement 403
3 ompe te included ngles. = I = 40 4 Stte nd justify you conclusion. Δ nd ΔGI e simil (SS). c 1 Re-dw Δ nd Δ JK so te ngles/sides coespond. c 6.6 110 6 6.6 K 110 30 10 40 J 10 ook t te side tios. 10 6.6 = nd = J 10 KJ 6.6 = 1 = 1 3 eck te included ngles, nd KJ. We do not know KJ. 4 Stte you conclusion. We e unle to sow tt te tingles e simil. d 1 Re-dw Δ nd ΔMNO so te ngles/sides coespond. d 6.6 110 6 N 30 10 40 M 110 0 O None of te tee tests (, SSS o SS) cn e pefomed since we know te mesuements of only one side nd one ngle in te ΔMNO. Stte tis in witing. We e unle to detemine wete te tingles e simil. eing le to detemine te coesponding ngles in tingle is vitl to ll tee tests on simility. In some cses te ctul vlue of one of te ngles is unknown, ut y mtemticl deduction its size cn e found. Tis equies tt you ememe you ngle popeties. Some of tese popeties e sown in te following tle. ngle desciption 1. Wen two lines intesect, tey fom veticlly opposite ngles, wic e equl.. Wen tnsvesl cuts pllel lines, nume of ngles e fomed. igm Tnsvesl Pllel lines 3. ltente ngles e on opposite (ltente) sides of te tnsvesl nd e lwys equl. 4. oesponding ngles e lwys equl. c c d d 5. o-inteio ngles lie on te sme side of te tnsvesl nd witin te pllel lines. Tey lwys dd up to 180. d c + = 180 c + d = 180 404 Mts Quest 11 Stndd Genel Mtemtics
WORK XMP 19 Sow tt Δ Δ. 3.75 7. 7.5 4.8 10 5 TUTORI eles-1389 Woked emple 19 TINK WRIT/RW 1 w te tingles septely wit ngles mked nd side mesuements sown. 7.5 4.8 c 11.5 10 15 7. c Stte te coesponding pis of ngles wic e equl in size. Specify te eson. = (sed) = (coesponding) = (coesponding) 3 Wite you conclusion. Δ Δ using test. If, in pi of simil tingles, te lengts of t lest one pi of coesponding sides e given, te tio (scle fcto) cn e estlised. It cn ten e used to find missing lengts in one of te tingles, povided tt te coesponding lengts in te ote tingle e known. WORK XMP 0 Find te vlue of te ponumel in te figue t igt. 7.5 10 15 TINK 1 stlis wete te tingles nd e simil y pplying te test. Since te tingles e simil, tei coesponding sides e in te sme tio. WRIT = (sed) = (coesponding) = (coesponding) Δ Δ ( test) 15 = 10 = 7.5 3 Solve fo. 15 = 10 7.5 15 7.5 = 10 = 11.5 Note: Wen foming n eqution, plce te unknown vlue in te numeto to mke clcultion steps esie o use S clculto to solve te eqution. PTR 10 Spe nd mesuement 405
ecise 10 Simil tingles 1 W18 Fo ec of te following, compe Δ nd ΔF nd stte wete tey e simil, o tee is not enoug infomtion given fo decision to e mde. Justify you nswes. 0 10 5.5 115 40 6 0 4 F 40 8.5 5 6 cm 40 3 cm 3 cm 6 cm 10 F c 5.65 3.375 7.875.5 3.5 1.5 F d 6 1 8 40 6 110 30 F lculte te size of te missing ngles. Justify you nswe. 4 c d c 68 c 3 c d e d 78 c d e e 99 y 63 f c e d 3 58 3 M Te mgnitude of ngle is: 30 180 30 60 180 60 70 4 M Te mgnitude of ngle is: 64 18 5 154 104 30 64 5 W19 Sow tt Δ is simil to Δ ecuse: te coesponding ngles e equl te coesponding side tios e equl (to vlue ote tn 1). 6 cm 5 cm 6 In ec of te following digms, find nd e-dw two tingles tt e simil. Give esons fo you nswe. c d e 4 cm 4. cm 3.5 cm 6.8 cm 406 Mts Quest 11 Stndd Genel Mtemtics
7 W0 Find te vlues of te ponumels (to 1 deciml plce). 37 cm.4 31 cm 9.6 y * y 9.6 cm * 6. cm 1.4 c y 5 18 1 d 6 15 5 y 4 e 1.7 m 4 m 3.4 m f cm 30 y 3 cm z 8 M Te side lengts of tingle e 18 mm, 4 mm nd 30 mm. Tingle F is simil to tingle nd its sotest side is 1 mm. Te peimete of te tingle F is: 48 mm 5 mm 64 mm 7 mm 108 mm 9 cm 9 t you 18t itdy pty you wnt to sow slides of you nd you fiends gowing up. Te digm t igt sows te set-up of te pojecto lens, slide nd pojecto sceen. If: () slide is 5 cm 5 cm () te distnce fom to is 10 cm (c) te pojecto sceen is 1.5 m 1.5 m ow f oizontlly fom te sceen do you need to plce te pojecto s lens so te imge just coves te entie sceen? 10I Symmety -dimensionl spe my e descied s ving eite line symmety nd/o ottionl symmety. ine symmety If it is possile to cut -dimensionl spe in suc wy tt it is divided into two mio imges, ten it possesses line symmety. Te line tt llows tis to occu is clled te is of symmety. Spes cn ve moe tn one is of symmety: Slide ens Pojecto sceen One is of symmety Two es of symmety Tee es of symmety Infinite nume of es of symmety n esy wy to ceck wete te line is n is of symmety of cetin spe is to fold te spe long tt line. If te two pts coincide (tt is, te spe folds onto itself), te line is n is of symmety. PTR 10 Spe nd mesuement 407
WORK XMP 1 Wic of te dotted lines in tis figue is n is of symmety? TINK 1 Visulise te tingle s eing folded long ec dotted line. If te tingle folds onto itself te line is n is of symmety. WRIT F TUTORI eles-1390 Woked emple 1 is n is of symmety. nd F e not. F is n is of symmety. Rottionl symmety If spe cn e otted out its cente so tt sits ectly on itself in less tn one complete evolution of 360, ten it is sid to ve ottionl symmety. Te nume of times spe sits on itself in one ottion is clled its ode of ottionl symmety. Fo emple, sque possesses ottionl symmety. y mking one cone nd otting te sque in clockwise diection ound point (its cente of ottion), it cn e seen tt in tun of 90, te sque ppes s it did oiginlly. = Oiginl Rotted on itself 90 Rotted on itself 180 Rotted on itself 70 Rotted on itself 360 Wit ec susequent tun of 90 te sque ppes s it oiginlly did. Witin te 360 evolution tee wee fou times te sque ppeed s its oiginl, so sque s ode of ottionl symmety is sid to e 4. n equiltel tingle s n ode of ottionl symmety of 3. 10 10 = 10 Te pinciples of line nd ottionl symmety cn lso e pplied to 3-dimensionl ojects. es of symmety n is of symmety is line out wic n oject cn otte so tt it ssumes positions identicl wit tose of te oiginl. ike ottionl symmety in -dimensionl spes, 3-dimensionl spe cn e descied y its ode of ottionl symmety. Tis is te nume of times te spe ssumes n ppence identicl wit tt in te oiginl position, witin one evolution of 360 out its is of symmety. sque-sed pymid s n ode of ottionl symmety of 4. In sque-sed pymid tee is only one is of symmety; oweve, ote 3-dimensionl ojects cn ve none o mny. 408 Mts Quest 11 Stndd Genel Mtemtics
Tesselltions Tee e mny emples in design, t nd citectue wee spes e used to cove sufce wit pttens wic leve no gps. Tese pttens e efeed to s tesselltions. Tesselltions cn e divided into tee min ctegoies: egul, omogeneous (o semi-egul) nd non-omogeneous (iegul). 1. In egul tesselltions, te ptten is mde y only one type of egul polygon.. In omogeneous tesselltions, te ptten is mde up of diffeent types of egul polygon. 3. In non-omogeneous tesselltions, ny type of spe cn e used. Te suject of tesselltions is lso pesented in pte 6 Geomety in two nd tee dimensions (pges 53 55). Non-omogeneous (iegul) tesselltions Tis ctegoy is not esticted to using egul polygons, so ny spes tt will fit togete to cove plne sufce cn e used. Te spes still need to fom definite ptten tt is epeted to cove te sufce. ecise 10I Symmety 1 W1 Wic of te dotted lines in ec of te figues sown is n is of symmety? G G F F c d G G F F opy te following figues into you wokook nd dw in ll te possile es of symmety. c d e f 3 w egul pentgon wit sides 1.5 cm in lengt nd intenl ngles ll ccutely mesuing 108. w in ll es of symmety. Te point wee tese lines intesect in egul polygon is known s te cente of ottion. Use tis point to otte te pentgon. c Toug wt ngle must te pentgon e otted efoe it supeimposes itself fo te fist time? (It is impotnt to mesue te ngle vey cefully to te neest degee.) d ow mny times does te pentgon supeimpose itself wen tuned 360? e Wt is its ode of ottion? PTR 10 Spe nd mesuement 409
4 Fo ec of te following spes, stte its ode of ottionl symmety. c d e f 5 Mke sque pymid y copying onto ppe o tin cdod, te net fo te sque pymid sown. ut ound te edge of te net, fold long te dotted lines nd ten stick te sides togete wit desive tpe. ow mny es of symmety does sque pymid ve? Wt is its ode of ottionl symmety? 6 ete tle wit te following edings. Veticl symmety oizontl symmety Rottionl symmety None etemine to wic column o columns ec of te cpitl lettes of te lpet (sown elow) elongs. FGIJKMNOPQRSTUVWXYZ 410 Mts Quest 11 Stndd Genel Mtemtics
Summy Pytgos teoem in two dimensions In ny igt-ngled tingle, te sque of te ypotenuse is equl to te sum of te sques of te ote two sides: c = + ypotenuse c Pytgos teoem in tee dimensions Use te following steps wen solving 3-dimensionl polems. Step 1. Identify te lengt tt needs to e found. Step. Identify te tingle tt contins te unknown lengt. Step 3. Re-dw tis tingle in two dimensions. If te tingle contining te unknown lengt s ote mesuements missing, find tese missing vlues fist fom ote tingles witin te figue. Unless stted in te polem, do not ound clcultions until te vey lst vlue is clculted nd ten give n nswe wit te ppopite nume of deciml plces. Peimete nd e Genelly: Unless specified, do not ound units until te end of te solution. If te mesuements in question include units, include tem in you finl nswe. Te peimete is te distnce ound closed -dimensionl spe. e efes to te sufce enclosed y te oundies of -dimensionl spe. Spe e Peimete Sque = P = 4 Rectngle = W P = ( + W) W Pllelogm = wee te eigt mesuement must e t igt ngle to te se mesuement. P = sum of ll sides Tpezium = 1 ( + ) wee te eigt mesuement must e t igt ngle to te se mesuement. P = sum of ll sides Romus y = 1 y P = sum of ll sides PTR 10 Spe nd mesuement 411
Spe e Peimete Tingle = 1 wee te eigt mesuement must e t igt ngle to te se mesuement. P = sum of ll sides Tingle icle c = s( s )( s )( s c) wee s = 1 ( + + c) (Use wen eigt mesuement is unknown.) = π P = + + c = π d o = π d n nnulus is te e etween two concentic cicles. e of nnulus: = π (R ), wee R is te dius of te lge cicle nd is te dius of te smlle cicle. secto is fomed y two dii of te cicle nd n c. θ e of secto = π 360 c lengt of secto = θ π 360 wee θ is te ngle of secto nd is te dius of te cicle. To find te e of composite spes, clculte te e of te individul spes fist. Totl sufce e (TS) Totl sufce e is te sum of te sufces of 3-dimensionl oject. Oject Net TS ue TS = 6 1 3 4 5 6 Rectngul pism w l w l l w TS = (w + lw + l) 41 Mts Quest 11 Stndd Genel Mtemtics
Oject Net TS ylinde TS = e of cicles + cuved sufce = π + π = π ( + ) π Spee Not sown TS = 4π one Sque-sed pymid S = Slnt eigt π S TS = e of se (cicle) + e of cuved sufce π = π + π S π S = π + π S = π ( + S) TS = e of sque + e of 4 tingles = + 4 1 = + Volume Fomuls fo ll types of ojects e not possile. Fo tose ojects witout fomul: 1. dw te net of te oject. wok out te diffeent spes tt mke up te net 3. clculte tei individul es 4. dd ll te individul pts togete. Te volume of n oject is te mount of spce tt te oject occupies. It is mesued in cuic units. Pisms e 3-dimensionl ojects wit unifom coss-sections nd pllel sides. Te coss-section is epesented y te ends of te pism. Te eigt is te dimension pependicul to te coss-sectionl e. Volume of pism = coss-sectionl e eigt of te pism Spe oss-sectionl spe Volume ylinde V = e of cicle eigt = π Tingul pism e = π e = 1 V = e of tingle eigt = 1 Note: owecse epesents te eigt of te tingle. PTR 10 Spe nd mesuement 413
Spe oss-sectionl spe Volume Rectngul pism V = e of ectngle eigt W = W W e = W ue e = V = e of sque eigt = = = 3 (since in sque, = ) Volume of pymid = 1 e of se pependicul eigt 3 Spe Flt end (se) spe Volume one V = 1 3 e of cicle eigt V = 1 π 3 Sque pymid V = 1 3 e of sque eigt V = 1 3 Rectngul pymid W V = 1 3 e of ectngle eigt = 1 W 3 W Tingul pymid V = 1 3 e of tingle eigt 1 1 V = 3 Note: owecse epesents te eigt of te tingle. o not confuse te lowecse in te fomul fo tingul pism/pymid o tpezoidl pism wit te uppecse. Uppecse = totl eigt of te pism o pymid, wile lowecse is te eigt of te tingle, pllelogm of tpezium. Fo spee: Volume = 4 π 3 3 414 Mts Quest 11 Stndd Genel Mtemtics
pcity Simil figues Simil tingles Symmety Te cpcity of contine is te mount tt it cn old. To find te fluid cpcity of contine, find its volume in cuic units fist nd ten convet, using te following ules: 1 cm 3 = 1 m, 1 m 3 = 1 k = 1000 Simil figues ve te sme spe ut diffeent sizes. Te coesponding ngles in ec simil figue e te sme; oweve, te coesponding side lengts diffe y scle fcto. Tingles e simil if: 1. ll coesponding ngles e equl (), o. ll coesponding sides e in te sme tio (SSS), o 3. two sides e in te sme tio nd te included ngles e equl (SS). Wen comping tingles it is est to ve tem dwn wit te sme oienttion. Wen foming n eqution to solve fo missing dimension, plce te unknown vlue in te numeto to mke te clcultion steps esie. Te line tt divides -dimensionl spe into two pts tt e mio imges of ec ote is clled te is of symmety. -dimensionl spe cn e clssified y te nume of lines of symmety it possesses nd te nume of times te spe supeimposes itself wen otted ound fied point, clled te ode of ottionl symmety. 3-dimensionl spe cn e clssified y te nume of plnes of symmety it possesses nd te nume of times te spe supeimposes itself wen otted ound te cosen is of symmety (its ode of ottionl symmety). PTR 10 Spe nd mesuement 415
pte eview MUTIP OI 1 mine te digm.te vlue of te ponumel to 1 deciml plce is: 6. 6.0 5.8 5.6 5.4 Te longest stick tt cn e plced inside cue wit side lengt of m s te lengt (to deciml plces) of: 3.46 m 3.44 m 3.50 m 3.48 m 3.5 m 3 Te peimete of ectngle wit n e of 56 cm nd one side of 8 cm is: 30 cm 7 cm 15 cm 8 cm 6 cm 4 Te e of one-mete wide gvel pt ound lwn 5 m 15 m is: 84 m 80 m 78 m 76 m 8 m 5 mine te digm. Te eigt of te spe sown is: 3.5 cm 0.5 cm 19.5 cm 15.5 cm 5.5 cm 1 9 8 cm 5 16 cm 6 Te e of te sded section in tis digm is: 1. cm 11.0 cm 141.4 cm 1414 mm 1110 mm 7 mm 7 Te TS of ectngul pism is 16 cm. If te se of te pism is 6 cm 5 cm, wt is its eigt?.5 cm 3 cm 3.4 cm 3.5 cm 4 cm 8 mine te digm t igt. Te TS of te solid sown is given y: 3π m 3π m 3 4 π m π m π m 5 9 mine te digm t igt. Te volume of te spe sown is given y: 1 3π m n + 4 3π m 3π m + π m n π m n 1 3π n(m + ) 1 3π m (n + m) 10 cylindicl contine of dius nd eigt is eing filled wit snd using conicl contine, lso of dius nd eigt. ow mny cones of snd e needed to fill te cylinde to cpcity? π 3π 3 1 3 11 ectngul swimming pool s cpcity of 1000 k. If te pool is 50 m long nd m deep, ow wide is it? 10 m 0 m 5 m 50 m 100 m 1 If te volume of contine is 1000 mm 3, wt is its cpcity in m? 0.1 m 1 m 10 m 100 m 1000 m 13 Te volumes of two simil-sped cylindes e 1000 cm 3 nd 3375 cm 3. In simplest fom, wt is te tio of tei sufce es? 10:15 4:9 4:6 :3 9:4 π 3 n m cm m 416 Mts Quest 11 Stndd Genel Mtemtics
14 On te citect s pln, ouse lock is 7.5 cm wide, wile te ctul (el-life) widt of te lock is 60 m. Wt is te tio scle of te pln? 1:80 8:1 1:8 1:800 80:1 15 Te lengt of is: 4 cm 16 cm 10 cm 8 cm 8 6 6 cm cm 1 16 Te lengts of nd y, espectively, e: 1 m nd 14 m 13 m nd 16 m 4 1 10 18 m nd 16 m 16 m nd 18 m 15 y 18 m nd 14 m 17 tee csts sdow of 10 m, wile you 0.45-m wooden stick csts sdow of 100 cm. Wt is te eigt of te tee? 45 m 4 m 4.5 m 5 m 5.5 m 18 Te nume of es of symmety of te figue sown is: 5 7 3 9 1 To llow people ccess to te gym, te mnge s decided to plce mp ove set of stis. Wt lengt of time (to te neest centimete) does e need to pucse? SORT NSWR Rmp 7 cm 16.5 cm -m sufod is to e plced in locke. Will it fit down te side sown y te line o will it ve to go digonlly coss s sown y te line? 3 Find (to 1 deciml plce): i te peimete ii te e of te following spes. c 6 cm 0 35' 8. cm m 15.67 cm 1.8 m 0.6 m 0.6 m 4 To secue te.4-m-ig poles of ec volleyll net you need to ttc guy ope to ing seven-eigts of te wy up ec pole. Te guy opes e.5 m in lengt. w digm of te sitution nd include ll dimensions given. ow f out (to 1 deciml plce) fom te se of te poles will te guy opes go? 5 designe vse s te spe of tuncted, sque-sed pymid. Te se of te vse is sque wit side lengt of 15 cm. Te e of te sque opening is 70.56 cm. c of te fou sides is tpezium wit slnt sides 9 cm long. Find (to te neest sque centimete) te totl sufce e of te vse. 6 If te volume of cone of eigt 10 cm is 61.8 cm 3, sow tt tis volume is incesed y fcto of 8 if te dimensions of te cone e douled. 7 cylindicl cn of dink olds 375 m of liquid. If te e of te se is 31.5 cm, find te eigt of te cn. PTR 10 Spe nd mesuement 417
8 Te spes elow ve eite een enlged o educed y scle fcto fom fied point. In ot cses, clculte y wt scle fcto te oiginl will need to e multiplied to poduce te imge. 1.5 cm 10 cm 1 cm 6 cm 5 cm 0.4 cm 8 cm Note: Te imge is te sded e. 9 Two ectngles e simil. Te lge ectngle s dimensions e 4 cm 1 cm. If te longest side on te smlle ectngle is 16 cm, clculte te ote dimension. 10 cicle wit dius of 1 cm s n e of 1385.4 cm nd note cicle s n e of 153.9 cm. Using te pinciples of tios, wt is te dius of tis cicle to te neest centimete? 11 Find te lengt of in te digm sown. 1 cm 1 opy te cessod t igt into you wokook nd dw in te lines of symmety. Rotte te od ound its cente of ottion (te point wee te lines of symmety coss). Wt is its ode of ottion? 16 cm XTN RSPONS 1 en nd le e enovting tei toom. Inspied y cetive eiit fom te ecent ome Sow, tey decided to use vious geometic spes in tei design. Te window wll is te fist section to e enovted. Te old window is to e eplced wit two new windows s sown elow nd ten te wll is to e tiled. 1. m.5 m Te window on te igt is in te spe of n equiltel tingle, wile te one on te left is in te spe of tpezium. Te sote pllel side of te tpezium is te sme lengt s te sides of te tingul window nd te longe pllel side is equl to.5 m. ot windows e to e 1. m ig. lculte: te peimete of ec window to te neest centimete te cost of wood (to te neest doll) needed to fme te windows, if te wood is sold t $13.50 pe mete c te e of ec window d te totl cost of te glss (to te neest doll), piced t $45 pe m. Te window wll is to e tiled wit glss tiles in te spe of omus s sown t igt. e Will te tiles tessellte? Give esons fo you nswe nd dw smll segment of te ptten to illustte it. f Smll spces etween te tiles e to e filled wit specil gout. Find te lengt (to te neest centimete) ound ec tile tt needs to e filled wit gout. g Find te e of ec tile. If te window wll is 4.6 m.7 m, find te totl e tt needs to e tiled. 30 cm i Te tiles e sold in oes of 1. Use te esults fom pts g nd to find te nume of oes tt en nd le need to pucse. dd n et 10% fo ekges to te nume of tiles needed. 0 cm 418 Mts Quest 11 Stndd Genel Mtemtics
fte successfully coping wit te window wll (see Question 1), le nd en decide to impove ote pts of tei toom. Fist tey wnt to decote ote wlls wit mosic fetues s sown t igt. Find te e of ec fetue to te neest cm. If en wnts 6 of ec type of fetue, find te totl e of te mosic needed; ten dd 10% et fo ekges nd cutting. c Mosic cn e pucsed in sque seets 30 cm 30 cm t $ ec. (c seet cn ten e cut s needed, ut only wole seets cn e pucsed.) Find te totl cost of te mosic fetues. Net le wnts to eplce te old mio wit new one s sown t igt. Te mio is in te spe of mjo segment, tt is, cicle wit section of it cut off stigt. (Te cut-off section will e complemented y smll glss selves.) d If ot te dius nd te lengt of te cut-off edge e 50 cm ec, find te totl e of te mio (to te neest cm ). e le wnts to ve te edges of te mio evelled. Tis cn e esily done t te locl glss stoe t cost of $0 pe mete. Find te cost of te evelling jo. Te most ding fetue of te new toom is te etemely moden looking sowe sceen. It cn e tougt of s section of te cuved sufce of cylinde 1.4 m long. Te cuved edge is in te spe of n c, sutended y 10 ngle fom te cente of cicle wit dius of 3 m. f Find te lengt of te cuved edge. g Find te totl e of te glss in te sowe sceen. el te uilde dvises en nd le tt since tey wnt fmeless sceen, it s to e mde fom specil tougened 10-mm-tick glss. Wt fction of cuic mete is te volume of glss in tis unusul sowe sceen? 3 cocolte mnufctue s ougt out new selection of cocolte spes. Tey e filled wit delicious oney/cocolte liquid. 3 one 1.5 60 1.5 Two tingul pisms joined togete uved edge 10 40 cm Mio = 3 5 cm 40 cm 40 cm uved edge 1.5 Spee 50 cm 50 cm 60 Selves 1.4 m 1.5.5 ue Rectngul pism ll mesuements given e in cm. Fo ec cocolte spe, clculte: i te TS of cocolte equied ii te volume of oney/cocolte liquid equied (to 1 deciml plce). 1.5 1.5 1.5 3 Tpezoidl pism 1.5 1.5 PTR 10 Spe nd mesuement 419
If o contins fou of ec type of cocolte spe, wt is te totl e of cocolte needed to mke one o? c ocolte comes in sque seets of 50 cm 50 cm. ow mny oes cn e poduced pe seet of cocolte? d ow mny millilites of oney/cocolte liquid e equied pe o? e If te liquid comes in one-lite contines, ow mny oes of cocolte cn e poduced pe lite? 4 Te scle dwing t igt is of gden sed. If te el-life widt of te sed is m, wt is te scle fcto 3 cm i in cm to m? (e) ii s tio :? iii s fction? Use tis scle fcto to clculte ll te el dimensions indicted y te ponumels. c If te sed is to e pinted, nd tis includes evey side ecept te oof (sown t igt): 3 cm i clculte te TS to e pinted Widt () ii ow mny lites of pint e equied if one lite coves 4 m? d Find te totl mount of spce ville inside te sed. Give te nswe in cuic metes to 1 deciml plce. e Te longest stick tt cn fit inside te sed will go fom te cone on te floo, lelled, to te opposite cone on top of te oof, lelled. Find te lengt of suc stick. 5 Te digm elow sows tingul coffee tle. (f) 3.75 cm.4 cm (c) 3 cm (d) 3.75 cm () Roof section (not pinted) O IGIT O doc-964 Test Youself pte 10 nlge te oject y scle fcto of. 10 cm Te oiginl design of te tletop is dwn s sown. If te longest side of te ctul tle is 1.6 m, wt is te 8 cm scle tio? c Find te ctul lengts of te ote two sides of te tletop. Te tletop is mde fom mle wit glss inset in te middle. Te glss inset is lso in te spe of te tingle wit side lengts 60 cm 48 cm 3 cm. d e te two tingles (tt is, te mle top nd te glss inset) simil? Justify you nswe. e oes te tletop ve: i line symmety? ii ottionl symmety? f esign tingul tletop wit tee lines of symmety nd n ode of ottionl symmety of 3. Specify te lengts of te sides of you tletop. g w te pln of you design fom pt f using scle tio of 1:0. 40 Mts Quest 11 Stndd Genel Mtemtics
IT ctivities pte opene IGIT O 10 Quick Questions doc-9608: (pge 379) 10 Pytgos teoem IGIT OS SkillST 10.1 doc-9609: (pge 380) doc-9610: (pge 380) 10 Pytgos teoem in tee dimensions IGIT O doc-9610: (pge 38) 10 Peimete nd e TUTORI W5 eles-1384: (pge 384) IGIT OS SkillST 10. doc-9611: (pge 386) SkillST 10.3 doc-961: (pge 386) 10 Totl sufce e (TS) TUTORI W 10 eles-1385: (pge 390) IGIT OS Investigtion doc-9613: (pge 39) WokST 10.1 doc-9614: (pge 39) 10 Volume TUTORI W 13 eles-1386: (pge 395) IGIT OS SkillST 10.4 doc-9615: (pge 396) SkillST 10.5 doc-9616: (pge 397) SkillST 10.6 doc-9617: (pge 397) Investigtion doc-9618: (pge 397) doc-9619: (pge 397) 10F pcity IGIT O doc-960: (pge 398) 10G Simil figues TUTORIS W 17 eles-1387: (pge 400) IGIT OS WokST 10. doc-961: (pge 40) Investigtion doc-96: (pge 40) 10 Simil tingles INTRTIVITY Simil tingle int-0811: (pge 40) TUTORI W19 eles-1389: (pge 405) 10I Symmety TUTORI W1 eles-1390: (pge 408) pte eview IGIT O Test Youself doc-964: (pge 40) To ccess eookpus ctivities, log on to www.jcplus.com.u PTR 10 Spe nd mesuement 41
nswes PTR 10 SP N MSURMNT ecise 10 Pytgos teoem 1 15 9. c 18.3 d 7.6 e 6.6 f 4.7 87.7 cm 0.88 m 3 = 0.7 m = 3.7 4 coss te pk: totl distnce = 8.98 km Vi te steets: totl distnce = 3.5 km Using te steets will enle te unne to complete 30 + km. 5 uoy 1 uoy S/F 100 m c 0 km ecise 10 Pytgos teoem in tee dimensions 1.4 5.3 c 14.8 d 7.9 1.5 3.1 c 18.1 d 18.5 3 5 cm 6.9 cm ecise 10 Peimete nd e 1 i 97.98 mm ii 48 mm i 1.4 m ii 4.88 m c i 168 cm ii 54 cm d i 3 cm ii 49.34 cm e i 8.75 m ii 13.40 m i 540 cm ii 106 cm i 60.73 m ii 45.71 m c i 9.74 cm ii 1.4 cm 3 c 4 i ii 1.1 m 1. m i 6 m m 1 m 1. m 3 m c 5.9 m 5 Rdius = 0.6 m o 60 cm 6 33% 7 30.6 cm 8 107 cm 9 Te cin is 137.4 cm long, so it s too sot. 10 474 cm 34.4% c 110 cm ii m ecise 10 Totl sufce e (TS) 1 577 cm 670.4 cm c 483.1 m d 47.0 m e 61. cm f 463.0 mm viii vi c ii d i e v f iii g iv i i vii 3 1 cm 11.4 m 4 5.0 m 7.0 m c 7.6 cm d 8.4 cm 5 No, te dimete of te cok is 1.6 cm, so it will fll toug. 6 7 8 9 6.3 m 10 Tingul pism c 15.3 cm 11 35.6 mm 1 6 48 cm ecise 10 Volume 1 ii vi c v d iii e iv f i 489.6 cm 3 848.7 cm 3 c 50.7 cm 3 d 1075. cm 3 e 768.9 cm 3 3 336 cm 3 80.5 cm 3 c 118 mm 3 d 750 cm 3 4 $54.00 5 6.1 m 3 13 717.1 cm 3 c 356. cm 3 6 179.59 cm 3 808.17 cm 3 c 69.4 cm 3 7 4. cm 8 Rectngul pism: 91 00 cm 3 omposite pism: 94 004.6 cm 3 Tpezium pism: 79 19 cm 3 ontine will e est s pples equie (51 160) 81 90 cm 3. ontine s too muc spce, wile c does not ve enoug. 9 3744 cm 3 ecise 10F pcity 1 750 m 0.8 c 500 m d 40 e 6 000 000 cm 3 6 000 000 6000 f 1 000 g 4. k 7.5 k 7500 i 5. cm 3 j 6000 cm 3 k 0 000 m = 0 000 cm 3 l 5.3 m 3 660 m 3 7.1 lites 4 63 mm 5 3 95 m 6 35 k i 9. minutes ii 41.7 minutes 7 60 cm ecise 10G Simil figues 1 nswes sown in popotion, ut not to size. ' ' O ' ' ' O F F' ' ' ' nswe sown in popotion, ut not to size. 3 Simil Simil c Not simil d Simil e Simil f Not simil g Not simil Simil 4 :5 o 4:10 :5 o 4 :10 4:5 o 16:100 4:5 e of smll ectngle = 8 cm 5 nswes will vy. 3: c 9:4 d nswes will vy. 6 6:10 o 3:5 6 3 :10 3 3 3 :5 3 16:1000 7:15 7:15 Volume of smlle cylinde 7 = 785.4 16 = 169.6 cm 3 7 8:7 16.7 cm 3 ecise 10 Simil tingles 1 Simil Simil c Simil d Not enoug infomtion,, c, d = 78 = 68 ; nd c = 11 c c = 3 ;,, d, e = 74 d,, e = 10 ; c, d = 78 e y = 36, = 63 f, e = 1 ; = 3 ; d, c = 6 3 4 5 nswes will vy. 6 Se sme ngle oesponding ngles oesponding ngles ' Se sme ngle oesponding ngles oesponding ngles 4 Mts Quest 11 Stndd Genel Mtemtics
c d e qul (veticlly opposite) ngles oesponding ngles oesponding ngles Sed ngle oesponding ngles oesponding ngles s sown 7 = 7.4, y = 48 y = 3.1, = 3.1 c = 1.5, y = 19.5 d = 7.5, y = 10 e = 4 f = y = 75, z = 4 8 9.9 m ecise 10I Symmety 1 ll, (,, F, G),, F c,, F d None c e 3 nd 0 1 3 d 1 f 1 3 4 5 6 5 Plnes of symmety: 4 es of symmety: 1 Ode of ottionl symmety: 4 6 Veticl symmety I M O T U V W X Y oizontl symmety I K O X PTR RVIW Rottionl symmety I O S X Z None F G J P Q R N MUTIP OI 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 SORT NSWR 1 95 cm Te od just fits digonlly (tt is, long te line ). 3 i P = 48.0 cm ii = 19.4 cm i P = 1.3 m ii = 11.5 m c i P = 48 cm ii = 173.7 cm 4 Pole Guy-ope.5 m.1 m 5 TS = 617 cm 6 nswes will vy. 7 1 cm 8.5 1 9 8 cm 10 7 cm 11 9.6 cm 1 ines of symmety.4 m 1.4 m d $143 e Yes y y Node plus y = 180 Te sum of ngles ound ec node dd to 360º. f P = 7 cm g = 300 cm 3 = 9.5 m i 9 oes nnulus: = 766 cm ; popelle-sped: = 838 cm = 10 586.4 cm c $64 d = 768 cm e $6.36 f.4 m g 3.388 m 0.033 88 m 3 3 I II III IV V VI i TS (cm ) 13.076 1.897 7.069 13.5 19.5 17.096 ii Volume (cm 3 ) 3.1.9 1.8 3.4 5.6 4.4 33.6 cm c 7 full oes d 84.8 m e 11 full oes 4 i 3 cm to m ii 3:00 iii 3 00 = m =.5 m c = 1.6 m d = m e = m f =.5 m c i.76 m ii 5.69 lites d 16.8 m 3 e 5.01 m 5 c 7 d 5 times e 5 4 c 6 d Infinite e None f None XTN RSPONS 1 Tingulw indow: P = 4.16 m; Tpezoidl window: P = 6.7 m $147 c Tingul window: = 0.834 m ; Tpezoidl window: =.334 m O 1 : 16 c 0.96 m d Not simil (does not pss te SSS test) e i No ii No f quiltel tingle; side lengt my vy g nswes will vy depending on nswe to pt f. PTR 10 Spe nd mesuement 43