MultilingualMultiDomain/MultiDomain/cross_domain_outputs/Maths/encoded/test.bpe.en

▁What ▁then ▁are ▁the ▁requ is ites ▁which ▁such ▁a ▁rule ▁ought ▁to ▁have ?
▁So , ▁the ▁value ▁of ▁x ▁when ▁y ▁= ▁3 ▁is ▁1.
▁Again , ▁we ▁transl ate ▁so ▁that ▁the ▁points ▁A ▁and ▁A ' ▁co inc ide .
▁Now , ▁we ▁want ▁to ▁find ▁out ▁how ▁many ▁nut s ▁the ▁c row ▁e ats ▁in ▁4 ▁hours .
▁Here , ▁in ▁order ▁to ▁st ress ▁that ▁the ▁understanding ▁of ▁the ▁Py th ag ore an ▁the ore m ▁does ▁not ▁involved ▁coordin ates , ▁we ▁look ▁at ▁those ▁Chapter ▁7 ▁argu ments ▁in ▁a ▁coordinate ▁free ▁plane .
▁Therefore , ▁H ans ol ▁we igh s ▁42 ▁kil ogram s .
▁Now , ▁let ' s ▁find ▁out ▁the ▁total ▁amount ▁Bruce ▁spent ▁on ▁mango es .
▁Not ice ▁also ▁that ▁any ▁section ▁of ▁such ▁a ▁sol id ▁by ▁a ▁plane ▁par alle l ▁to ▁the ▁original ▁plan es ▁is ▁a ▁copy ▁of ▁the ▁original ▁plan ar ▁figure .
▁Finally , ▁let ' s ▁find ▁out ▁how ▁much ▁more ▁the ▁tail or ▁cut ▁off ▁the ▁sk irt ▁and ▁j ack et ▁combined ▁than ▁the ▁p ants ▁and ▁bl ouse ▁combined :
▁Now , ▁let ' s ▁count ▁the ▁number ▁of ▁p airs ▁that ▁sum ▁to ▁9 :
▁If ▁9 ▁so cc er ▁games ▁are ▁played ▁each ▁month ▁and ▁the ▁season ▁last s ▁for ▁3 ▁months , ▁then ▁the ▁total ▁number ▁of ▁so cc er ▁games ▁played ▁during ▁the ▁season ▁would ▁be :
▁Therefore , ▁we ▁cannot ▁determine ▁the ▁number ▁of ▁students ▁in ▁section ▁B ▁based ▁on ▁the ▁information ▁given .
▁9 ▁shel ves ▁* ▁6 ▁col oring ▁books ▁per ▁she lf ▁= ▁54 ▁col oring ▁books
▁Sol ve ▁real - wor ld ▁and ▁mathematical ▁problems ▁involving ▁area , ▁volume ▁and ▁surface ▁area ▁of ▁two - ▁and ▁three - ▁d imens ional ▁objects ▁composed ▁of ▁triangles , ▁qu adr ilater als , ▁poly g ons , ▁cub es , ▁and ▁right ▁pr ism s .
▁Now , ▁we ▁know ▁that ▁there ▁are ▁11 ▁pregnant ▁female ▁c ows .
▁Not ice ▁that ▁if ▁a ▁den ot es ▁the ▁first ▁term , ▁and ▁b ▁the ▁second ▁term ▁of ▁the ▁root , ▁the ▁first ▁complete ▁divis or ▁is ▁3 a 2 ▁+ ▁3 ab ▁+ ▁b 2 ▁, ▁and ▁the ▁second ▁trial - d iv is or ▁is ▁3 ( a ▁+ ▁b ) 2 ▁, ▁that ▁is , ▁3 a 2 ▁+ ▁6 ab ▁+ ▁3 b 2 ▁.
▁S av ings ▁per ▁b age l ▁= ▁Cost ▁per ▁b age l ▁ind iv id ually ▁- ▁Cost ▁per ▁b age l ▁in ▁a ▁doz en
▁The ▁goal ▁here ▁is ▁that ▁students ▁understand ▁the ▁process ▁to ▁calculate ▁dist ances : ▁this ▁process ▁invol ves ▁right ▁angles ▁and ▁the ▁Py th ag ore an ▁the ore m ▁and ▁students ▁are ▁to ▁understand ▁that ▁involve ment .
▁Therefore , ▁Su z an ne ▁will ▁need ▁13 ▁lo aves ▁of ▁bread ▁to ▁make ▁f rench ▁to ast ▁over ▁52 ▁weeks .
▁- ▁In gr id ▁was ▁16 5 ▁minutes ▁late .
▁For ▁any ▁poly gon al ▁figure ▁in ▁one ▁of ▁the ▁plan es , ▁swe ep ▁it ▁through ▁space ▁in ▁a ▁direction ▁per pend ic ular ▁to ▁the ▁starting ▁plane ▁until ▁we ▁reach ▁the ▁end ing ▁plane .
▁In ▁Figure ▁12 ▁are ▁three ▁3 D ▁figures ▁obtained ▁by ▁swe eping ▁a ▁figure ▁in ▁the ▁bott om ▁plane ▁out ▁to ▁the ▁top ▁plane .
▁St art ▁at ▁the ▁plan ar ▁figure ▁at ▁the ▁base ▁of ▁a ▁pr ism : ▁its ▁area ▁is ▁approxim ated ▁by ▁covering ▁the ▁the ▁figure ▁with ▁a ▁gr id ▁of ▁squ ares , ▁and ▁count ing ▁the ▁number ▁of ▁squ ares ▁inside ▁the ▁figure .
▁5 ▁hours ▁per ▁week ▁/ ▁5 ▁days ▁per ▁week ▁= ▁1 ▁hour ▁per ▁day
▁Since ▁2 .25 x ▁40 = 9 0, ▁they ▁will ▁needs ▁90 ▁bags ▁of ▁sand ▁for ▁the ▁large ▁sw ing ▁set .
▁In ▁section ▁1 ▁we ▁begin ▁by ▁gather ing ▁together ▁facts ▁about ▁angles ▁and ▁triangles ▁that ▁have ▁already ▁been ▁discussed ▁in ▁previous ▁gr ades .
▁To ▁find ▁out ▁how ▁many ▁photos ▁were ▁in ▁each ▁al b um , ▁we ▁divide ▁the ▁total ▁number ▁of ▁photos ▁by ▁the ▁number ▁of ▁al b ums .
▁F ra ction ▁of ▁col or ▁film s ▁= ▁30 y ▁/ ▁(31 y )
▁We ▁know ▁that ▁40 ▁bags ▁co vers ▁27 ▁cub ic ▁feet .
▁When ▁we ▁have ▁a ▁set ▁of ▁things ▁such ▁as ▁lengths ▁which ▁are ▁meas ur able ▁in ▁terms ▁of ▁any ▁one ▁of ▁them , ▁we ▁say ▁that ▁they ▁are ▁quant ities ▁of ▁the ▁same ▁kind .
▁Since ▁all ▁of ▁the ▁col or ▁film s ▁are ▁selected , ▁there ▁are ▁6 y ▁col or ▁film s ▁selected .
▁We ▁now ▁calculate ▁the ▁components ▁of ▁the ▁Py th ag ore an ▁formula :
▁4. ▁P ip er ▁returns ▁8 ▁b ars ▁for ▁her ▁ve gan ▁friends :
▁Paul ▁arrived ▁at ▁8 : 25 ▁a . m ., ▁so ▁S as ha ▁arrived ▁at ▁8 : 25 ▁a . m . ▁+ ▁65 ▁minutes ▁= ▁9 : 30 ▁a . m .
▁The ▁line ▁segment ▁where ▁two ▁faces ▁come ▁together ▁is ▁called ▁an ▁ed ge ▁of ▁the ▁poly hed ron .
▁68 .4 0 15 ▁inches ▁/ ▁12 ▁inches / f oot ▁= ▁5 .7 00 12 5 ▁feet
▁The ▁store ▁started ▁with ▁40 ▁col oring ▁books ▁and ▁sold ▁20 ▁of ▁them , ▁so ▁they ▁had ▁40 ▁- ▁20 ▁= ▁20 ▁col oring ▁books ▁left .
▁Also , ▁since ▁3 a 2 ▁b ▁+ ▁3 ab 2 ▁+ ▁b 3 ▁= ▁(3 a 2 ▁+ ▁3 ab ▁+ ▁b 2 ▁) b , ▁the ▁complete ▁divis or ▁is ▁obtained ▁by ▁adding ▁3 ab ▁+ ▁b 2 ▁to ▁the ▁trial ▁divis or ▁3 a 2.
▁Therefore ▁the ▁total ▁area ▁is ▁32 ( 8 8 +2 7 .5 + 20 ) ▁= ▁135 .5 ▁s q .
▁All en ▁read ▁10 ▁pages ▁a ▁day , ▁so ▁in ▁5 ▁days , ▁he ▁read :
▁Man ufact ure rs ▁sell ▁break fast ▁ce re als ▁by ▁volume ▁or ▁weight ▁( us ually ▁25 ▁o z .) ▁but ▁determine ▁the ▁size ▁of ▁cont ain ers ▁on ▁economic ▁and ▁a est het ic ▁ground s .
▁Not ice ▁that ▁the ▁shape ▁of ▁the ▁package ▁with ▁the ▁smallest ▁surface ▁area ▁is ▁the ▁package ▁that ▁most ▁closely ▁rese mb les ▁a ▁cub e ▁and ▁the ▁package ▁with ▁the ▁greatest ▁surface ▁area ▁is ▁the ▁package ▁that ▁is ▁least ▁like ▁a ▁cub e .
▁Reg ular ▁season ▁games ▁= ▁( 50 ▁teams ▁* ▁49 ▁games ▁per ▁team ) ▁/ ▁2 ▁= ▁24 50 ▁/ ▁2 ▁= ▁12 25 ▁games
▁If ▁so , ▁we ▁are ▁in ▁the ▁case ▁of ▁Figure ▁6 a .
▁Therefore , ▁the ▁original ▁number ▁of ▁but ter f lies ▁Jer ry ▁had ▁is ▁7 4.
▁So , ▁there ▁are ▁160 ▁three - d ig it ▁numer als , ▁1, 600 ▁four - d ig it ▁numer als , ▁and ▁16 ,000 ▁five - d ig it ▁numer als ▁that ▁meet ▁the ▁criteria .
▁P it ts b ur gh ▁to ▁Prov idence : ▁Four ▁and ▁an ▁e igh th ▁inches , ▁or ▁4 12 .5 ▁miles ;
▁Since ▁we ▁know ▁how ▁many ▁pages ▁P ra k ash ▁can ▁copy ▁in ▁1 ▁hour , ▁we ▁can ▁find ▁out ▁how ▁many ▁pages ▁Sub ash ▁can ▁copy ▁in ▁1 ▁hour ▁by ▁subtract ing ▁P ra k ash ' s ▁rate ▁from ▁their ▁combined ▁rate :
▁- ▁Each ▁field ▁goal ▁is ▁worth ▁3 ▁points , ▁so ▁2 ▁field ▁goals ▁would ▁be ▁2 ▁* ▁3 ▁= ▁6 ▁points .
▁When ▁you ▁multiply ▁three ▁by ▁nine ▁you ▁get ▁twenty - se ven .
▁and ▁whose ▁hypot en use ▁has ▁length ▁230 ▁m .
▁2 ▁rect angular ▁faces ▁( the ▁side ▁faces ▁in ▁the ▁diagram ) ▁that ▁are ▁15 x ▁30 ▁each ▁contrib ute ▁450 ▁s q .
▁27 ▁boxes ▁÷ ▁9 ▁boxes ▁per ▁stop ▁= ▁3 ▁st ops
▁Since ▁the ▁total ▁work ▁is ▁W , ▁we ▁have :
▁Hence , ▁if ▁a ▁given ▁number ▁cont ain ▁a ▁decimal , ▁we ▁divide ▁it ▁into ▁groups ▁of ▁two ▁figures ▁each , ▁by ▁beginning ▁at ▁the ▁decimal ▁point ▁and ▁mark ing ▁toward ▁the ▁left ▁for ▁the ▁integral ▁number , ▁and ▁toward ▁the ▁right ▁for ▁the ▁decimal .
▁Since ▁A ▁is ▁5 8, ▁the ▁total ▁runs ▁after ▁10 ▁innings ▁is ▁10 ▁* ▁58 ▁= ▁5 80 .
▁Students ▁should ▁experiment ▁with ▁this ▁statement ▁with ▁a ▁vari ety ▁of ▁rect angles .
▁Furthermore , ▁put ▁the ▁a pe x ▁of ▁the ▁con e ▁directly ▁over ▁one ▁of ▁the ▁vert ices ▁of ▁the ▁base ▁square .
▁If ▁you ▁know ▁the ▁area ▁of ▁a ▁shape , ▁can ▁you ▁determine ▁that ▁distance ▁around ▁the ▁shape ?
▁Since ▁the ▁tra ck ▁is ▁one - qu ar ter ▁m ile ▁long ▁and ▁she ▁circ led ▁it ▁12 ▁times , ▁she ▁covered :
▁We ▁know ▁that ▁after ▁giving ▁away ▁the ▁to w els , ▁she ▁ended ▁up ▁with ▁19 ▁to w els , ▁so :
▁We ▁look ▁at ▁the ▁collection ▁of ▁triangles ▁with ▁two ▁side ▁lengths ▁a ▁and ▁b ▁with ▁a \\\\\\\\ ge q ▁b .
▁The ▁total ▁number ▁of ▁d uc ks ▁from ▁both ▁po nds ▁is ▁30 ▁( from ▁the ▁smaller ▁po nd ) ▁+ ▁D ▁( from ▁the ▁larger ▁po nd ).
▁Figure ▁11 ▁is ▁that ▁of ▁an ▁is os ce les ▁right ▁triangle , ▁D elt a ▁ABC , ▁l ying ▁on ▁top ▁of ▁a ▁square .
▁First , ▁let ' s ▁find ▁out ▁how ▁much ▁the ▁tail or ▁cut ▁off ▁the ▁sk irt ▁and ▁j ack et ▁combined :
▁So , ▁15 ▁divided ▁by ▁9 ▁leaves ▁a ▁remainder ▁of ▁6.
▁The ▁original ▁cost ▁price ▁of ▁the ▁car ▁( C P ) ▁= ▁Rs . ▁52 94 1 .1 76 47 0 58 8 24
▁16 00 ▁gallons / day ▁needed ▁/ ▁400 ▁gallons / day ▁produced ▁= ▁4 ▁days
▁This ▁means ▁that ▁each ▁cat ▁cat ches ▁one ▁m ouse ▁in ▁that ▁certain ▁amount ▁of ▁time .
▁In ▁common ▁language , ▁a ▁plane ▁section ▁is ▁also ▁called ▁cross ▁section .
▁We ▁want ▁to ▁establish ▁this ▁formula ▁to ▁comp ute ▁the ▁volume ▁of ▁pr ism s :
▁Two ▁figures ▁are ▁said ▁to ▁be ▁similar ▁if ▁there ▁is ▁a ▁sequ ence ▁of ▁rigid ▁mot ions ▁and ▁dil ations ▁that ▁takes ▁one ▁figure ▁on to ▁the ▁other .
▁When ▁infl ated , ▁the ▁false ▁Great ▁Py ram id ▁was ▁22 5 ▁m ▁high , ▁with ▁a ▁sl ant ▁height ▁of ▁230 .5 ▁m ▁for ▁any ▁one ▁of ▁the ▁triangle ▁faces ▁( by ▁sl ant ▁height , ▁we ▁mean ▁the ▁distance ▁from ▁its ▁base ▁to ▁the ▁a pe x ▁of ▁the ▁py ram id ▁in ▁the ▁plane ▁of ▁the ▁triangle .)
▁The ▁amount ▁of ▁silver ▁in ▁the ▁first ▁solution ▁is ▁( x / 100 ) ▁* ▁5 ▁liters .
▁A ▁bought ▁the ▁b icy cle ▁for ▁Rs . ▁150 ▁and ▁made ▁a ▁20% ▁profit ▁on ▁it .
▁Add ing ▁the ▁gr adu ates ▁themselves , ▁we ▁have :
▁The ▁next ▁requ is ite ▁is ▁that ▁minute ▁adjust ments ▁of ▁the ▁rule ▁should ▁be ▁so ▁made ▁as ▁to ▁allow ▁of ▁the ▁simpl est ▁possible ▁statements ▁of ▁the ▁laws ▁of ▁nature .
▁7 ▁p iz z as ▁left ▁+ ▁4 ▁p iz z as ▁eaten ▁= ▁11 ▁p iz z as
▁Now , ▁we ▁need ▁to ▁check ▁which ▁pair ▁gives ▁a ▁sum ▁that ▁is ▁at ▁least ▁36 .
▁S olution ▁1 ▁( v ol ume ): ▁3 ▁inches ▁is ▁1/ 4 =0 .25 ▁feet , ▁so ▁the ▁volume ▁of ▁sand ▁that ▁was ▁used ▁is ▁0 .25 x ▁9 x ▁12 =2 7 ▁cub ic ▁feet .
▁Cal c ulate ▁the ▁length ▁of ▁P Q ▁using ▁the ▁Py th ag ore an ▁the ore m .
▁Now , ▁to ▁find ▁the ▁distance ▁between ▁the ▁two ▁train s ▁after ▁10 ▁hours , ▁we ▁subtract ▁the ▁distance ▁travel ed ▁by ▁the ▁first ▁train ▁from ▁the ▁distance ▁travel ed ▁by ▁the ▁second ▁train :
▁So , ▁180 ▁photos ▁÷ ▁9 ▁al b ums ▁= ▁20 ▁photos ▁per ▁al b um .
▁To ▁find ▁out ▁her ▁rank ▁from ▁the ▁bott om , ▁we ▁subtract ▁the ▁number ▁of ▁students ▁a head ▁of ▁her ▁from ▁the ▁total ▁number ▁of ▁students ▁and ▁add ▁1 ▁( for ▁J aya ▁herself ).
▁To ▁calculate ▁the ▁amount ▁of ▁Ch lor ine , ▁Ben z ene , ▁and ▁B rom ine ▁required ▁to ▁re act ▁with ▁M eth ane ▁to ▁form ▁D ich lor om eth ane , ▁Ch lor ob en z ene , ▁and ▁B rom om eth ane , ▁we ▁need ▁to ▁consider ▁each ▁re action ▁separ ately ▁and ▁balance ▁the ▁chem ical ▁equations .
▁The ▁same ▁method ▁will ▁apply ▁to ▁numbers ▁of ▁more ▁than ▁two ▁groups ▁of ▁figures , ▁by ▁considering ▁in ▁each ▁case ▁a , ▁the ▁part ▁of ▁the ▁root ▁already ▁found , ▁as ▁so ▁many ▁tens ▁with ▁respect ▁to ▁the ▁next ▁figure ▁of ▁the ▁root .
▁When ▁AC ▁is ▁vertical , ▁we ▁have ▁the ▁right ▁triangle , ▁for ▁which ▁the ▁length ▁c = s q r t { a ^{2} + b ^{2} } .
▁Therefore , ▁the ▁probability ▁that ▁the ▁computer ▁company ▁will ▁not ▁get ▁the ▁soft w are ▁contract ▁is ▁approximately ▁0.5 55 6 ▁or ▁55 .5 6 %.
▁Students ▁taking ▁only ▁music ▁= ▁40 ▁- ▁10 ▁= ▁30
▁2. ▁If ▁the ▁py ram id ▁is ▁cut ▁with ▁a ▁plane ▁pass ing ▁through ▁the ▁top ▁ver te x ▁and ▁per pend ic ular ▁to ▁the ▁base , ▁the ▁inter section ▁of ▁the ▁py ram id ▁and ▁the ▁plane ▁is ▁a ▁tri angular ▁cross - section .
▁Since ▁all ▁three ▁train s ▁are ▁running ▁towards ▁each ▁other ▁on ▁par alle l ▁lines , ▁we ▁need ▁to ▁consider ▁the ▁maximum ▁relative ▁speed ▁for ▁the ▁time ▁calculation .
▁To ▁find ▁the ▁cost ▁per ▁b age l ▁when ▁buy ing ▁a ▁doz en , ▁we ▁divide ▁the ▁total ▁cost ▁by ▁the ▁number ▁of ▁b age ls ▁in ▁a ▁doz en :
▁R ate ▁= ▁1 ▁/ ▁18 ▁( K ir an ▁comple t es ▁1/ 18 ▁of ▁the ▁work ▁per ▁day ▁for ▁the ▁remaining ▁work )
▁The ▁perimeter ▁of ▁the ▁box ▁of ▁side ▁length ▁30 ▁cm ▁is ▁120 ▁cm .
▁We ▁can ▁express ▁the ▁sum ▁of ▁these ▁integ ers ▁as :
▁The ▁trad er ▁has ▁an ▁overall ▁loss ▁of ▁approximately ▁1.6 84 % ▁on ▁the ▁whole ▁transaction .
▁If ▁not , ▁after ▁reflection ▁in ▁any ▁side ▁of ▁triangle ▁ABC ▁put s ▁us ▁in ▁the ▁case ▁of ▁Figure ▁7.
▁We ▁therefore ▁put ▁0 ▁for ▁the ▁next ▁figure ▁of ▁the ▁root .
▁Qu ant ity ▁of ▁mango es ▁= ▁4 95 ▁/ ▁55 ▁per ▁kg
▁Let ' s ▁call ▁the ▁integer ▁we ▁are ▁multiply ing ▁by ▁" n ".
▁12 ▁inv it ations ▁÷ ▁3 ▁inv it ations ▁per ▁pack ▁= ▁4 ▁pack s
▁Note ▁that ▁a ▁cub e ▁is ▁a ▁special ▁kind ▁of ▁rect angular ▁pr ism , ▁one ▁in ▁which ▁all ▁ed ges ▁are ▁of ▁the ▁same ▁length .
▁1. ▁T ae h y ung ▁has ▁fewer ▁apples ▁than ▁N am j oon .
▁It ▁is ▁eas ier ▁and ▁more ▁fun ▁to ▁experiment ▁with ▁this ▁at ▁webs ites ▁set ▁up ▁for ▁this ▁purpose .
▁Therefore , ▁before ▁today , ▁they ▁had ▁rep av ed ▁4 13 3 ▁inches ▁of ▁the ▁road .
▁The ▁che f ▁used ▁a ▁total ▁of ▁5 ▁pot at oes ▁for ▁the ▁lunch ▁c row d ▁and ▁2 ▁pot at oes ▁for ▁the ▁d in ner ▁c row d .
▁F ind ▁the ▁square ▁root ▁of ▁25 x 2 ▁− ▁20 x 3 ▁y ▁+ ▁4 x 4 ▁y ▁2 ▁.
▁Now , ▁we ▁know ▁that ▁this ▁$ 400 ▁in ▁interest ▁is ▁approximately ▁6 .5% ▁of ▁the ▁bond ' s ▁selling ▁price .
▁Let ' s ▁look ▁at ▁pr ism s ▁and ▁con es , ▁as ▁they ▁are ▁sol id ▁sh ap es ▁created ▁by ▁drawing ▁out ▁a ▁plan ar ▁poly gon .
▁So , ▁27 ▁so cc er ▁games ▁were ▁played ▁in ▁total ▁during ▁the ▁season .
▁The ▁weight ▁already ▁on ▁the ▁f ence ▁is ▁18 ▁l bs , ▁so ▁the ▁remaining ▁weight ▁the ▁f ence ▁can ▁hold ▁is ▁20 ▁- ▁18 ▁= ▁2 ▁l bs .
▁Next , ▁let ' s ▁calculate ▁the ▁score ▁for ▁the ▁E ag les :
▁So , ▁the ▁average ▁weight ▁increases ▁by ▁2.5 ▁kg ▁when ▁the ▁new ▁person ▁we igh ing ▁95 ▁kg ▁repl aces ▁the ▁person ▁we igh ing ▁75 ▁kg .
▁Third ▁week : ▁4 ▁inst ances ▁of ▁using ▁phone ▁* ▁3 ▁dem er its ▁per ▁inst ance ▁= ▁12 ▁dem er its
▁1. ▁For ▁the ▁loan ▁to ▁Paul :
▁54 ▁remaining ▁col oring ▁books ▁+ ▁33 ▁sold ▁col oring ▁books ▁= ▁87 ▁col oring ▁books
▁The ▁largest ▁such ▁triangle ▁is ▁obtained ▁when ▁the ▁plane ▁of ▁the ▁section ▁includes ▁three ▁vert ices ▁of ▁the ▁cub e .
▁18 ▁pages ▁/ ▁2 ▁hours ▁= ▁9 ▁pages / h our
▁The ▁smallest ▁two - d ig it ▁number ▁divis ible ▁by ▁8 ▁is ▁16 ▁( which ▁is ▁8 * 2 ), ▁and ▁the ▁largest ▁two - d ig it ▁number ▁divis ible ▁by ▁8 ▁is ▁96 ▁( which ▁is ▁8 * 12 ).
▁Therefore , ▁the ▁score ▁in ▁the ▁11 th ▁in ning ▁is ▁64 9 ▁- ▁5 80 ▁= ▁6 9.
▁Since , ▁in ▁7 th ▁grade , ▁we ▁consider ▁only ▁this ▁case , ▁we ▁will ▁not ▁use ▁the ▁ad ject ive ▁right .
▁So ▁the ▁weight ▁he ▁can ▁lif t ▁in ▁the ▁military ▁press ▁is ▁2 C ▁p ounds .
▁- ▁D il an ▁was ▁15 5 ▁minutes ▁late .
▁What ▁is ▁the ▁distance ▁from ▁one ▁corn er ▁of ▁the ▁floor ▁( point ▁A ▁in ▁the ▁figure ) ▁to ▁the ▁opposite ▁corn er ▁on ▁the ▁ce iling ?
▁If ▁they ▁can ▁copy ▁300 ▁pages ▁in ▁40 ▁hours ▁together , ▁then ▁in ▁1 ▁hour ▁they ▁can ▁copy :
▁Then , ▁he ▁let ▁ele ven ▁go , ▁so ▁he ▁had ▁1 .25 x ▁- ▁11 ▁but ter f lies ▁left .
▁Let ' s ▁list ▁these ▁facts ▁about ▁similar ity :
▁In ▁a ▁20 ▁over ▁match , ▁there ▁are ▁20 ▁overs ▁x ▁6 ▁balls ▁per ▁over ▁= ▁120 ▁balls .
▁Of ten ▁the ▁name ▁cross ▁section ▁is ▁reserv ed ▁for ▁a ▁section ▁of ▁a ▁3 D ▁object ▁that ▁is ▁par alle l ▁to ▁a ▁particular ▁plane ▁of ▁sy m m et ry ▁of ▁the ▁object , ▁or ▁per pend ic ular ▁to ▁a ▁particular ▁line ▁of ▁sy m m et ry .
▁Therefore , ▁the ▁limit ing ▁factor ▁is ▁the ▁spreading ▁time , ▁not ▁the ▁production ▁time .
▁12 ▁( f ull ▁capacity ) ▁- ▁8 ▁( c urrent ▁number ▁of ▁apples ) ▁= ▁4 ▁apples .
▁By ▁recon f ig uring ▁the ▁picture ▁as ▁in ▁Figure ▁9, ▁we ▁can ▁show ▁that ▁this ▁is ▁also ▁a ^{2} + b ^{2} .
▁Thus ▁what ▁is ▁outside ▁those ▁four ▁triangles ▁must ▁have ▁area ▁c ^{2} .
▁If , ▁therefore , ▁an ▁integral ▁square ▁number ▁is ▁divided ▁into ▁groups ▁of ▁two ▁figures ▁each , ▁from ▁the ▁right ▁to ▁the ▁left , ▁the ▁number ▁of ▁figures ▁in ▁the ▁root ▁will ▁be ▁equal ▁to ▁the ▁number ▁of ▁groups ▁of ▁figures .
▁Per cent age ▁of ▁eth an ol ▁= ▁(4 .75 ▁liters ▁/ ▁47 .5 ▁liters ) ▁* ▁100
▁To ▁solve ▁for ▁P ( S ), ▁we ▁re ar range ▁the ▁equation :
▁M iles ▁brought ▁4 ▁kg ▁of ▁flour , ▁so :
▁The ▁d imens ions ▁of ▁the ▁faces , ▁in ▁this ▁case , ▁are ▁11 ▁x ▁8, ▁11 ▁x ▁2.5 , ▁and ▁8 ▁x ▁2.5 ▁s q .
▁Now , ▁when ▁3 y ▁is ▁divided ▁by ▁9, ▁the ▁remainder ▁is ▁the ▁same ▁as ▁the ▁remainder ▁when ▁6 ▁is ▁divided ▁by ▁9, ▁which ▁is ▁6.
▁L ength s ▁are ▁meas ured ▁by ▁the ▁foot - ru le .
▁The ▁ca f eter ia ▁used ▁7 ▁p ies ▁* ▁8 ▁apples ▁per ▁pie ▁= ▁56 ▁apples ▁to ▁make ▁p ies .
▁The ▁same ▁method ▁may ▁be ▁applied ▁to ▁longer ▁expressions ▁by ▁considering ▁a ▁in ▁the ▁typ ical ▁form ▁3 a 2 ▁+ ▁3 ab ▁+ ▁b 2 ▁to ▁represent ▁at ▁each ▁stage ▁of ▁the ▁process ▁the ▁part ▁of ▁the ▁root ▁already ▁found .
▁If ▁we ▁proceed ▁to ▁tr ig on omet ry ▁this ▁cor rel ation ▁receiv es ▁a ▁more ▁exact ▁determ ination ▁in ▁the ▁famil iar ▁shape ▁sin ▁B ▁sin ▁C ▁sin ▁A ▁= ▁= ▁, ▁a ▁b ▁c ▁a 2 ▁= ▁b 2 ▁+ ▁c 2 ▁− ▁2 b c ▁cos ▁A , ▁with ▁two ▁similar ▁form ul æ .
▁Let ' s ▁denote ▁the ▁amount ▁of ▁bird se ed ▁a ▁par ake et ▁e ats ▁per ▁day ▁as ▁P ▁grams .
▁Let ▁c ▁be ▁the ▁length ▁of ▁the ▁hypot en use .
▁By ▁the ▁Py th ag ore an ▁the ore m ▁h ^{2} + 50 ^{2} =2 30 ^{2} .
▁Each ▁par ake et ▁e ats ▁2 ▁grams ▁of ▁bird se ed ▁per ▁day .
▁The ▁total ▁profit ▁for ▁the ▁month ▁is ▁the ▁mean ▁daily ▁profit ▁multip lied ▁by ▁the ▁number ▁of ▁days ▁in ▁the ▁month :
▁So , ▁the ▁original ▁purch ase ▁prices ▁before ▁tax ▁are ▁approximately :
▁To ▁find ▁out ▁how ▁many ▁past ries ▁he ▁still ▁has , ▁we ▁subtract ▁the ▁number ▁of ▁past ries ▁sold ▁from ▁the ▁original ▁number ▁of ▁past ries ▁made :
▁In ▁Chapter ▁7 ▁we ▁used ▁these ▁facts ▁to ▁calculate ▁the ▁area ▁of ▁the ▁t il ted ▁square , ▁which ▁is ▁c ^{2} , ▁where ▁c ▁is ▁the ▁length ▁of ▁the ▁hypot en use ▁of ▁the ▁four ▁triangles .
▁This ▁figure ▁consists ▁of ▁two ▁rect angles ▁and ▁4 ▁triangles .
▁Let ' s ▁follow ▁through ▁on ▁this ▁with ▁another ▁example : ▁Among ▁all ▁rect angles ▁of ▁perimeter ▁50 ▁units , ▁find ▁the ▁rectangle ▁with ▁the ▁largest ▁area .
▁We ek ▁3: ▁Alex ' s ▁collection ▁dou bles ▁again , ▁so ▁she ▁has ▁8 ▁* ▁2 ▁= ▁16 ▁pens .
▁Total ▁b ars ▁P ip er ▁has ▁to ▁distrib ute ▁= ▁200 ▁- ▁8 ▁= ▁19 2 ▁b ars
▁3. ▁Sol ve ▁real - wor ld ▁and ▁mathematical ▁problems ▁involving ▁volume ▁and ▁surface ▁area ▁of ▁three - d imens ional ▁objects ▁composed ▁of ▁triangles , ▁qu adr ilater als , ▁poly g ons , ▁cub es , ▁and ▁right ▁pr ism s .
▁Therefore , ▁J ane ▁exerc ises ▁1 ▁hour ▁per ▁day .
▁Now ▁the ▁circ les ▁are ▁concent ric .
▁In ▁Chapter ▁7 ▁we ▁observed ▁that ▁this ▁is ▁a ▁square ; ▁now , ▁let ▁us ▁check ▁that ▁this ▁is ▁so .
▁The ▁result ▁of ▁multiply ing ▁37 95 42 1 ▁by ▁863 4 .25 ▁is ▁3 2, 77 4, 67 0, 5 42 .2 5.
▁After ▁the ▁11 th ▁in ning , ▁his ▁average ▁increases ▁by ▁1, ▁making ▁it ▁A ▁+ ▁1.
▁When ▁infl ated , ▁the ▁false ▁Great ▁Py ram id ▁had ▁a ▁square ▁base ▁of ▁side ▁length ▁100 ▁m .
▁M ike ▁gave ▁you ▁22 ▁pens .
▁When ▁a ▁larger ▁cub e ▁is ▁pain ted ▁and ▁then ▁cut ▁into ▁smaller ▁cub es , ▁the ▁smaller ▁cub es ▁that ▁end ▁up ▁being ▁pain ted ▁on ▁only ▁two ▁faces ▁are ▁the ▁ones ▁that ▁come ▁from ▁the ▁ed ges ▁of ▁the ▁larger ▁cub e , ▁but ▁not ▁the ▁corn ers .
▁First ▁week : ▁6 ▁inst ances ▁of ▁being ▁late ▁* ▁2 ▁dem er its ▁per ▁inst ance ▁= ▁12 ▁dem er its
▁To ▁find ▁the ▁area ▁of ▁a ▁triangle : ▁design ate ▁one ▁its ▁sides ▁as ▁the ▁base , ▁and ▁denote ▁its ▁length ▁by ▁b .
▁To ▁find ▁15% ▁of ▁8 0, ▁you ▁can ▁multiply ▁80 ▁by ▁0 .1 5 ▁( since ▁15% ▁is ▁the ▁same ▁as ▁15 / 100 ▁or ▁0 .1 5 ).
▁The ▁poly g ons ▁that ▁make ▁up ▁the ▁outer ▁surface ▁of ▁the ▁poly hed ron ▁are ▁called ▁the ▁faces ▁of ▁the ▁poly hed ron .
▁This ▁is ▁because ▁rigid ▁mot ions ▁do ▁not ▁change ▁lengths , ▁and ▁dil ations ▁change ▁all ▁lengths ▁by ▁the ▁same ▁number , ▁the ▁factor ▁of ▁the ▁d ilation .
▁This ▁means ▁that ▁the ▁effective ▁selling ▁price ▁is ▁136 % ▁of ▁the ▁cost ▁price :
▁Inte rest ▁rate ▁( r ) ▁= ▁4 % ▁per ▁ann um ▁= ▁0 .04 ▁( in ▁decimal )
▁4 80 ▁o un ces ▁- ▁(5 ▁p ounds ▁* ▁16 ▁o un ces / p ound ) ▁= ▁4 80 ▁o un ces ▁- ▁80 ▁o un ces ▁= ▁400 ▁o un ces
▁The ▁rect angular ▁area ▁under ▁the ▁small ▁sw ing ▁set ▁measures ▁9 ▁feet ▁by ▁12 ▁feet ▁and ▁required ▁40 ▁bags ▁of ▁sand ▁to ▁increase ▁the ▁dep th ▁of ▁3 ▁inches .
▁Now ▁we ▁have ▁the ▁length ▁of ▁the ▁rectangle ▁(20 ▁units ) ▁and ▁the ▁radi us ▁of ▁the ▁circle ▁( 50 ▁units ).
▁49 38 ▁inches ▁( total ▁rep av ed ▁so ▁far ) ▁- ▁80 5 ▁inches ▁( rep av ed ▁today ) ▁= ▁4 13 3 ▁inches ▁( rep av ed ▁before ▁today )
▁0.9 5 ▁meters ▁* ▁39 .37 ▁inches / m eter ▁= ▁37 .4 0 15 ▁inches
▁Since ▁we ▁can ' t ▁have ▁a ▁fraction ▁of ▁a ▁bar , ▁let ' s ▁assume ▁they ▁each ▁take ▁33 ▁b ars ▁and ▁we ' ll ▁deal ▁with ▁the ▁remainder ▁later .
▁If ▁we ▁now ▁reflect ▁in ▁that ▁short ▁leg , ▁we ▁land ▁right ▁on ▁triangle ▁IV .
▁So ▁that ▁the ▁complete ▁divis or , ▁3 a 2 ▁+ ▁3 ab ▁+ ▁b 2 ▁, ▁will ▁be ▁represented ▁by ▁3 ( x ▁+ ▁y ) 2 ▁+ ▁3 ( x ▁+ ▁y ) z ▁+ ▁z ▁2 ▁.
▁The ▁25 ▁sub tra cted ▁is ▁really ▁2500 , ▁that ▁is , ▁a 2 ▁, ▁and ▁the ▁complete ▁divis or ▁2 a ▁+ ▁b ▁is ▁2 ▁× ▁50 ▁+ ▁7 ▁= ▁10 7.
▁So , ▁in ▁calcul ating ▁area ▁we ▁will ▁make ▁a ▁choice ▁of ▁face ▁to ▁serve ▁as ▁base ; ▁in ▁most ▁cases ▁it ▁doesn ' t ▁matter , ▁and ▁often ▁the ▁context ▁direct s ▁us ▁to ▁a ▁proper ▁choice ▁of ▁base .
▁B ren ' s ▁current ▁age ▁= ▁8 x ▁= ▁8 ▁* ▁5 ▁= ▁40 ▁years
▁In ▁seventh ▁grade , ▁students ▁saw ▁this ▁to ▁be ▁true ▁by ▁drawing ▁an ▁ar bit rary ▁triangle , ▁cutting ▁out ▁the ▁angles ▁at ▁the ▁vert ices , ▁and ▁putting ▁them ▁at ▁the ▁same ▁ver te x .
▁Therefore , ▁the ▁fraction ▁of ▁the ▁radi us ▁of ▁the ▁circle ▁that ▁rep res ents ▁the ▁length ▁of ▁the ▁rectangle ▁is ▁2 / 5.
▁5 ▁minutes ▁( co ff ee ) ▁+ ▁12 ▁minutes ▁( reg ular ▁employ ee ▁status ) ▁+ ▁9 ▁minutes ▁( s en ior ▁employ ee ▁status ) ▁+ ▁12 ▁minutes ▁( over time ▁pay ro ll ) ▁+ ▁5 ▁minutes ▁( non - over time ▁pay ro ll ) ▁+ ▁10 ▁minutes ▁( em ails ) ▁+ ▁7 ▁minutes ▁( t ask ▁allocation ) ▁= ▁60 ▁minutes
▁Since ▁any ▁two ▁angles ▁of ▁the ▁same ▁measure ▁are ▁con g ru ent , ▁we ▁can ▁find ▁a ▁sequ ence ▁of ▁rigid ▁mot ions ▁that ▁takes ▁C A B ▁on to ▁C ' A ' B '.
▁In ▁summary , ▁to ▁re act ▁with ▁3 ▁m oles ▁of ▁water ▁and ▁form ▁3 ▁m oles ▁of ▁calc ium ▁hy dro x ide ▁and ▁an ▁equal ▁amount ▁of ▁calc ium ▁ph os ph ate , ▁we ▁will ▁need :
▁S olution ▁2 ▁( scale ▁factor ): ▁Since ▁we ▁have ▁to ▁multiply ▁both ▁the ▁length ▁and ▁the ▁wid th ▁by ▁1.5 , ▁the ▁area ▁that ▁needs ▁to ▁be ▁covered ▁is ▁1.5 ^{2} =2 .25 ▁times ▁as ▁large .
▁Its ▁sides ▁and ▁angles ▁are ▁inter rel ated — the ▁greater ▁angle ▁is ▁opposite ▁to ▁the ▁greater ▁side , ▁and ▁the ▁base ▁angles ▁of ▁an ▁is os ce les ▁triangle ▁are ▁equal .
▁By ▁what ▁factor ▁does ▁the ▁length ▁of ▁the ▁dia gon al ▁change , ▁and ▁by ▁what ▁factor ▁does ▁the ▁area ▁change ?
▁The ▁tail or ▁bought ▁90 ▁green ▁but t ons .
▁The ▁first ▁sol id ▁in ▁Figure ▁14 ▁is ▁a ▁rect angular ▁py ram id ▁because ▁its ▁base ▁is ▁a ▁rectangle ▁( pro b ably ▁a ▁square ) ▁and ▁the ▁last ▁sol id ▁is ▁a ▁tri angular ▁py ram id ▁because ▁its ▁base ▁is ▁a ▁triangle .
▁So , ▁S oph ia ▁has ▁yet ▁to ▁read ▁90 .00 00 00 00 00 000 3 ▁pages .
▁Use ▁Figure ▁22 ▁to ▁follow ▁along .
▁If ▁Roh an ▁sa ves ▁Rs . ▁2500 ▁at ▁the ▁end ▁of ▁the ▁month , ▁this ▁amount ▁rep res ents ▁the ▁remaining ▁percentage ▁of ▁his ▁salary ▁after ▁all ▁the ▁expenses ▁have ▁been ▁ded u cted .
▁St art ▁with ▁the ▁square ▁of ▁side ▁length ▁c ▁( so ▁of ▁area ▁c ^{2} ), ▁and ▁draw ▁the ▁right ▁triangles ▁of ▁leg ▁lengths ▁a ▁and ▁b , ▁with ▁hypot en use ▁a ▁side ▁of ▁this ▁square , ▁as ▁shown ▁in ▁the ▁top ▁left ▁of ▁Figure ▁10.
▁Rem aining ▁work ▁= ▁M ▁men ▁× ▁26 ▁days
▁is ▁such ▁a ▁site ▁in ▁which ▁area ▁and ▁perimeter ▁are ▁inter act ively ▁exp lo red ▁with ▁triangles .
▁- ▁Reg ular ▁employees : ▁6 ▁employees ▁x ▁2 ▁minutes ▁each ▁= ▁12 ▁minutes
▁Each ▁bird ▁we igh s ▁3 ▁l bs , ▁so ▁the ▁total ▁weight ▁of ▁the ▁6 ▁birds ▁is ▁6 ▁* ▁3 ▁= ▁18 ▁l bs .
▁They ▁put ▁4 ▁col oring ▁books ▁on ▁each ▁she lf , ▁so ▁to ▁find ▁out ▁how ▁many ▁shel ves ▁they ▁used , ▁you ▁divide ▁the ▁number ▁of ▁col oring ▁books ▁left ▁by ▁the ▁number ▁of ▁col oring ▁books ▁per ▁she lf :
▁This ▁great ▁subject ▁is ▁often ▁mis nam ed ▁as ▁An aly tical ▁Con ic ▁Sect ions , ▁thereby ▁f ix ing ▁attention ▁on ▁mere ly ▁one ▁of ▁its ▁sub d iv isions .
▁The ▁argument ▁is ▁not ▁fully ▁completed , ▁for ▁the ▁conf ig uration ▁of ▁Figure ▁6 a ▁is ▁not ▁the ▁only ▁possibility .
▁By ▁ext ension , ▁the ▁Greeks ▁concluded ▁that ▁the ▁volume ▁of ▁a ▁circular ▁con e ▁is ▁1/3 ▁the ▁volume ▁of ▁its ▁circum sc rib ed ▁cyl inder ▁( st ud ents ▁will ▁return ▁to ▁this ▁in ▁8 th ▁grade .
▁Thus , ▁if ▁0 .11 ▁be ▁the ▁cub e ▁root ▁of ▁a ▁number , ▁the ▁number ▁is ▁0 .11 ▁× ▁0 .11 x 0 .11 ▁= ▁0 .00 13 3 1 .
▁Thus , ▁if ▁the ▁part ▁of ▁the ▁root ▁already ▁found ▁is ▁x ▁+ ▁y , ▁then ▁3 a 2 ▁of ▁the ▁typ ical ▁form ▁will ▁be ▁represented ▁by ▁3 ( x ▁+ ▁y ) 2 ▁ ; ▁and ▁if ▁the ▁third ▁term ▁of ▁the ▁root ▁be ▁+ z , ▁then ▁3 ab ▁+ ▁b 2 ▁will ▁be ▁represented ▁by ▁3 ( x ▁+ ▁y ) z ▁+ ▁z ▁2 ▁.
▁For ▁the ▁other ▁two ▁lengths , ▁use ▁the ▁slope ▁triangles ▁as ▁shown ▁and ▁perform ▁the ▁same ▁calculation :
▁Number ▁of ▁cakes ▁bought ▁= ▁169 ▁- ▁32
▁If ▁the ▁square ▁root ▁of ▁a ▁number ▁has ▁decimal ▁places , ▁the ▁number ▁itself ▁will ▁have ▁twice ▁as ▁many .
▁P aint able ▁Are a ▁of ▁W all ▁A ▁= ▁6 m 2 ▁- ▁1 m 2 ▁= ▁5 m 2
▁Inte rest ▁in ▁5 ▁years ▁= ▁1200 ▁- ▁800 ▁= ▁Rs . ▁400
▁and ▁the ▁height ▁of ▁one ▁of ▁the ▁side ▁triangles ▁was ▁230 ▁m .
▁50 ▁allow able ▁dem er its ▁- ▁10 ▁dem er its ▁( per form ance ▁evaluation ▁ded uction ) ▁= ▁40 ▁allow able ▁dem er its
▁Se lling ▁Pr ice ▁of ▁the ▁first ▁car ▁( SP 1) ▁= ▁C P 1 ▁+ ▁(1 3 % ▁of ▁C P 1)
▁2 ▁tri angular ▁b ases , ▁each ▁of ▁which ▁is ▁10 8 ▁s q .
▁However , ▁this ▁is ▁not ▁true ▁in ▁the ▁conf ig uration ▁of ▁Figure ▁7.
▁The ▁base ▁a ▁square ▁with ▁each ▁side ▁100 ▁m ▁in ▁length .
▁If ▁we ▁calculate ▁the ▁areas ▁of ▁many ▁rect angles ▁of ▁the ▁same ▁perimeter , ▁we ▁see ▁that ▁the ▁if ▁the ▁side ▁lengths ▁are ▁cl oser ▁in ▁measure ▁then ▁the ▁area ▁of ▁the ▁rectangle ▁is ▁larger .
▁Car rie ▁brought ▁18 .9 0 4 ▁o un ces ▁more ▁flour ▁than ▁the ▁combined ▁flour ▁brought ▁by ▁M iles ▁and ▁Kat y .
▁Thus ▁the ▁first ▁triangle ▁lands ▁on ▁top ▁of ▁the ▁second ▁triangle , ▁and ▁so ▁they ▁are ▁similar ,
▁Total ▁work ▁= ▁30 ▁men ▁× ▁40 ▁days
▁If ▁the ▁base ▁has ▁a ▁center , ▁and ▁the ▁line ▁from ▁the ▁center ▁to ▁the ▁a pe x ▁is ▁per pend ic ular ▁to ▁the ▁base , ▁we ▁call ▁the ▁sol id ▁a ▁right ▁py ram id .
▁Since ▁the ▁only ▁change ▁in ▁the ▁total ▁weight ▁comes ▁from ▁repl ac ing ▁the ▁92 ▁kg ▁person ▁with ▁the ▁new ▁person , ▁the ▁increase ▁in ▁total ▁weight ▁is ▁the ▁weight ▁of ▁the ▁new ▁person ▁min us ▁92 ▁kg .
▁So , ▁it ▁follows ▁that ▁g am ma = 90 degrees .
▁Now ▁we ▁know ▁the ▁product ▁of ▁x ▁and ▁y ▁is ▁5 000 .
▁We ▁will ▁summar ize ▁this ▁development ▁in ▁the ▁following ▁sequ ence ▁of ▁ex amples .
▁Next , ▁we ▁calculate ▁the ▁gain ▁percent ▁using ▁the ▁formula :
▁So , ▁together ▁they ▁j ump ed ▁a ▁total ▁of ▁approximately ▁5 .7 00 12 5 ▁feet .
▁R ot ate ▁a ▁copy ▁of ▁the ▁triangle ▁around ▁the ▁ver te x ▁B , ▁and ▁then ▁transl ate ▁the ▁new ▁triangle ▁up wards ▁to ▁get ▁the ▁result ▁shown ▁in ▁Figure ▁3, ▁in ▁which ▁the ▁triangle ▁with ▁d as hed ▁sides ▁is ▁the ▁new ▁position ▁of ▁the ▁cop ied ▁triangle .
▁Now , ▁we ▁know ▁the ▁time ▁taken ▁to ▁cross ▁each ▁other ▁is ▁9 .9 9 92 000 63 99 4 8 8 ▁seconds .
▁What ▁are ▁the ▁possible ▁vol um es ▁of ▁these ▁pack ages ?
▁- ▁(2 , ▁7 ) ▁= ▁9 ▁( This ▁pair ▁sum s ▁to ▁9 )
▁The ▁total ▁number ▁of ▁p ies ▁eaten ▁is ▁27 .
▁Con stru ct ▁a ▁plane ▁that ▁contains ▁a ▁point ▁near ▁a ▁ver te x ▁( other ▁than ▁ver te x ▁A ) ▁on ▁one ▁of ▁the ▁three ▁ed ges , ▁a ▁point ▁in ▁the ▁middle ▁of ▁another ▁one ▁of ▁the ▁ed ges , ▁and ▁a ▁third ▁point ▁that ▁is ▁neither ▁in ▁the ▁middle ▁nor ▁co inc iding ▁with ▁the ▁first ▁point .
▁A ▁r ing ▁made ▁of ▁gold ▁that ▁has ▁a ▁diam eter ▁of ▁30 ▁cm ▁is ▁put ▁in ▁a ▁silver ▁display ▁box ▁so ▁that ▁the ▁r ing ▁just ▁f its .
▁If ▁B aker ▁initially ▁had ▁169 ▁cakes ▁and ▁still ▁has ▁32 ▁cakes ▁left , ▁then ▁his ▁friend ▁bought ▁the ▁difference ▁between ▁the ▁two ▁numbers .
▁We ▁know ▁that ▁after ▁some ▁operation ▁on ▁2 x ▁+ ▁9, ▁the ▁result ▁is ▁5 1, ▁and ▁we ▁also ▁know ▁that ▁the ▁original ▁number ▁x ▁is ▁4.
▁This ▁means ▁that ▁after ▁14 ▁days , ▁there ▁are ▁still :
▁Now ▁not ▁every ▁sol id ▁shape ▁is ▁made ▁out ▁of ▁cub es , ▁but ▁if ▁we ▁take ▁the ▁unit ▁cub e ▁small ▁enough , ▁this ▁method ▁will ▁produce ▁a ▁good ▁approxim ation ▁of ▁the ▁volume .
▁If ▁three ▁numbers ▁are ▁in ▁geomet r ical ▁progress ion , ▁the ▁middle ▁number ▁is ▁called ▁the ▁geomet r ical ▁mean ▁of ▁the ▁other ▁two ▁numbers .
▁Here ▁we ▁are ▁given ▁the ▁d imens ions ▁V =3 2 40 ▁c u .
▁Out ▁of ▁these ▁rot ten ▁apples , ▁70% ▁of ▁them ▁sme lled .
▁What ▁is ▁the ▁length ▁of ▁the ▁r ing ▁of ▁gold ?
▁When ▁the ▁deal er ▁allow s ▁25 ▁art icles ▁to ▁be ▁sold ▁at ▁the ▁cost ▁price ▁of ▁20 ▁art icles , ▁the ▁effective ▁selling ▁price ▁per ▁article ▁for ▁this ▁deal ▁is ▁the ▁cost ▁price ▁of ▁20 ▁art icles ▁divided ▁by ▁25 ▁art icles , ▁which ▁is :
▁V ol ume ▁is ▁found ▁by ▁multiply ing ▁the ▁area ▁of ▁the ▁base ▁( is os ce les ▁triangle ) ▁by ▁the ▁length ▁of ▁the ▁pr ism : ▁V = B l .
▁We ▁have ▁two ▁solutions ▁for ▁k : ▁0 ▁and ▁2.
▁We ▁are ▁given ▁P ( H ▁ ∩ ▁S ), ▁which ▁is ▁the ▁probability ▁of ▁getting ▁both ▁contract s :
▁In ▁the ▁accomp any ing ▁m ap ▁of ▁the ▁n ort he ast ▁United ▁States , ▁one ▁in ch ▁rep res ents ▁100 ▁m ies .
▁Since ▁3 /5 ▁of ▁the ▁people ▁on ▁the ▁bus ▁were ▁Dutch , ▁we ▁calculate :
▁What ▁is ▁the ▁length ▁of ▁its ▁hypot en use ?
▁This ▁situation ▁is ▁illust rated ▁in ▁Figure ▁14 , ▁where ▁the ▁triangles ▁have ▁been ▁moved ▁by ▁rigid ▁mot ions ▁so ▁that ▁they ▁have ▁legs ▁that ▁are ▁horizontal ▁and ▁vertical , ▁and ▁they ▁have ▁the ▁ver te x ▁A ▁in ▁common .
▁50 ▁minutes / day ▁* ▁14 ▁days ▁= ▁700 ▁minutes
▁A ▁he x ag on al ▁pr ism ▁has ▁a ▁he x ag on ▁at ▁both ▁the ▁top ▁and ▁bott om , ▁and ▁a ▁he x ag on ▁has ▁6 ▁vert ices .
▁So , ▁Sam antha ' s ▁last ▁name ▁has ▁7 ▁letters , ▁Bob b ie ' s ▁last ▁name ▁has ▁10 ▁letters , ▁and ▁Alex ' s ▁last ▁name ▁has ▁12 ▁letters .
▁Com b ined ▁at omic ▁weight ▁of ▁the ▁two ▁at oms ▁of ▁the ▁other ▁element ▁= ▁15 9 .9 2 ▁am u
▁Peter ▁needs ▁enough ▁bird se ed ▁to ▁last ▁a ▁week ▁(7 ▁days ), ▁so ▁the ▁total ▁amount ▁of ▁bird se ed ▁needed ▁for ▁a ▁week ▁is ▁7 ▁* ▁(3 P ▁+ ▁28 ▁+ ▁2 P ) ▁= ▁26 6 ▁grams .
▁The ▁mole c ular ▁weight ▁of ▁a ▁single ▁mole cu le ▁of ▁H 2 O ▁( w ater ) ▁is ▁not ▁7 2.
▁Rec all ▁that ▁in ▁chapter ▁2, ▁in ▁the ▁section ▁on ▁the ▁slope ▁of ▁a ▁line , ▁we ▁introduced ▁the ▁idea ▁of ▁a ▁d ilation ▁in ▁order ▁to ▁show ▁that ▁the ▁slope ▁of ▁a ▁line ▁can ▁be ▁calculated ▁using ▁any ▁two ▁points .
▁D ▁= ▁(1 / 3) ▁* ▁(18 / 1) ▁( In ver ting ▁the ▁divis or ▁and ▁multiply ing )
▁V er ify ▁that ▁dil ations ▁take ▁lines ▁into ▁lines , ▁takes ▁par alle l ▁lines ▁to ▁par alle l ▁lines ▁and ▁that ▁a ▁line ▁and ▁its ▁image ▁under ▁a ▁d ilation ▁are ▁par alle l .
▁This ▁is ▁because ▁the ▁" sequ ence ▁of ▁rigid ▁mot ions ▁and ▁dil ations " ▁need ▁not ▁include ▁any ▁dil ations , ▁in ▁which ▁case ▁it ▁exhib its ▁a ▁con g ru ence .
▁What ▁is ▁the ▁length ▁of ▁the ▁dia gon al ▁of ▁a ▁rectangle ▁of ▁side ▁lengths ▁1 ▁in ch ▁and ▁4 ▁inches ?
▁Next , ▁let ' s ▁find ▁out ▁how ▁much ▁the ▁tail or ▁cut ▁off ▁the ▁p ants ▁and ▁bl ouse ▁combined :
▁The ▁distance ▁from ▁the ▁base ▁to ▁the ▁opp osing ▁ver te x ▁is ▁called ▁the ▁height ▁of ▁the ▁triangle ▁( see ▁Figure ▁2 ).
▁Now , ▁we ▁know ▁the ▁total ▁time ▁for ▁the ▁rel ay ▁race ▁was ▁200 ▁seconds .
▁If ▁the ▁long ▁side ▁has ▁length ▁20 ▁units , ▁the ▁short ▁side ▁has ▁length ▁5 ▁units ▁and ▁the ▁area ▁is ▁100 ▁square ▁units .
▁We ▁calculate " ▁6 .4 ^{2} = 40 .9 6, ▁so ▁it ▁makes ▁good ▁sense ▁to ▁use ▁the ▁value ▁6 .4 ▁to ▁approxim ate ▁s q r t { 4 1 } .
▁So , ▁Car ly ▁sent ▁200 ▁job ▁app lications ▁to ▁compan ies ▁in ▁her ▁state .
▁Thus , ▁if ▁0 .2 1 ▁is ▁the ▁square ▁root ▁of ▁some ▁number , ▁this ▁number ▁will ▁be ▁(0 .2 1) 2 ▁= ▁0 .2 1 × 0 .2 1 ▁= ▁0 .04 4 1, ▁and ▁if ▁0 .1 11 ▁be ▁the ▁root , ▁the ▁number ▁will ▁be ▁(0 .11 1) 2 ▁= ▁0 .1 11 x 0 .1 11 ▁= ▁0 .01 23 21 .
▁This ▁means ▁the ▁pass ing ▁marks ▁for ▁P ap er ▁A ▁are ▁60 ▁marks ▁( sec ured ) ▁+ ▁20 ▁marks ▁( ne ed ed ▁to ▁pass ) ▁= ▁80 ▁marks .
▁Distance _ A ▁= ▁50 ▁km / h r ▁× ▁8 ▁h r
▁We ▁have ▁two ▁argu ments ▁to ▁show ▁why ▁it ▁is ▁true .
▁Therefore , ▁we ▁can ▁re w rit e ▁3 y ▁as :
▁Note ▁that ▁each ▁length ▁has ▁been ▁multip lied ▁by ▁3, ▁while ▁the ▁area ▁of ▁the ▁larger ▁rectangle ▁is ▁9 ▁times ▁the ▁area ▁of ▁the ▁of ▁the ▁original ▁rectangle .
▁So , ▁the ▁length ▁of ▁the ▁cloth ▁in ▁the ▁first ▁sc en ari o ▁was ▁180 ▁meters .
▁Here ▁is ▁a ▁construction ▁that ▁comes ▁close .
▁Each ▁gr adu ate ▁can ▁bring ▁2 ▁parents , ▁so ▁for ▁50 ▁gr adu ates , ▁that ▁would ▁be :
▁The ▁a ero ▁flight ▁te es ▁come ▁in ▁p airs , ▁so ▁each ▁package ▁contains ▁2 ▁te es .
▁Thus , ▁I ▁and ▁IV ▁are ▁also ▁con g ru ent .
▁So , ▁the ▁tail or ▁cut ▁off ▁5 / 24 ▁in ch ▁more ▁from ▁the ▁sk irt ▁and ▁j ack et ▁combined ▁than ▁from ▁the ▁p ants ▁and ▁bl ouse ▁combined .
▁Now ▁recon f igure ▁these ▁triangles ▁and ▁the ▁inter ior ▁square ▁as ▁in ▁the ▁image ▁on ▁the ▁right .
▁Cost ▁per ▁b age l ▁in ▁a ▁doz en ▁= ▁ € 3 .1 66 7 ▁( rounded ▁to ▁four ▁decimal ▁places )
▁- ▁30% ▁of ▁inc and es cent ▁b ul bs ▁are ▁sw it ched ▁on , ▁which ▁is ▁0 .30 I .
▁If ▁J ane ▁exerc ises ▁for ▁8 ▁weeks ▁and ▁her ▁total ▁exerc ise ▁time ▁is ▁40 ▁hours , ▁we ▁can ▁calculate ▁the ▁number ▁of ▁hours ▁she ▁exerc ises ▁per ▁week ▁by ▁div iding ▁the ▁total ▁hours ▁by ▁the ▁number ▁of ▁weeks :
▁What ▁is ▁the ▁length ▁of ▁the ▁frame ▁of ▁silver ?
▁Since ▁the ▁comp ound ▁consists ▁of ▁one ▁at om ▁of ▁calc ium , ▁we ▁subtract ▁the ▁at omic ▁weight ▁of ▁calc ium ▁from ▁the ▁total ▁mole c ular ▁weight ▁of ▁the ▁comp ound ▁to ▁find ▁the ▁combined ▁at omic ▁weight ▁of ▁the ▁two ▁at oms ▁of ▁the ▁other ▁element .
▁So ▁far , ▁we ▁have ▁accounted ▁for ▁17 0 ▁ch airs .
▁Com b ining ▁these , ▁we ▁get ▁the ▁total ▁ratio ▁for ▁the ▁s port ▁form ulation ▁as :
▁Pr ice ▁after ▁additional ▁50% ▁staff ▁disc ount ▁= ▁0.4 5 d ▁- ▁( 50 % ▁of ▁0.4 5 d )
▁The ▁volume ▁of ▁the ▁box ▁is ▁the ▁product ▁of ▁its ▁side ▁length : ▁V ▁= ▁10 ▁x ▁10 ▁x ▁6 ▁= ▁600 ▁c u .
▁A ▁py ram id ▁has ▁been ▁described ▁as ▁the ▁sol id ▁formed ▁by ▁the ▁ag g reg ate ▁of ▁line ▁se gments ▁jo ining ▁points ▁on ▁a ▁given ▁poly gon al ▁figure ▁( the ▁base ) ▁in ▁a ▁plane ▁to ▁a ▁single ▁point ▁A ▁( the ▁a pe x ).
▁If ▁r > 1, ▁everything ▁exp and s ▁away ▁from ▁the ▁center ▁C , ▁and ▁if ▁r < 1 ▁everything ▁contract s ▁toward ▁C .
▁If ▁the ▁triangle ▁is ▁a ▁right ▁triangle , ▁we ▁can ▁design ate ▁one ▁leg ▁to ▁be ▁the ▁base ▁( of ▁length ▁b ) ▁and ▁the ▁other ▁to ▁be ▁the ▁height ▁( of ▁length ▁a ).
▁Her ▁daughters ▁sp lit ▁1 ▁sl ice , ▁so ▁that ' s ▁an ▁additional ▁1 ▁sl ice .
▁Then , ▁the ▁volume ▁of ▁the ▁py ram id ▁is
▁But ▁now , ▁by ▁red raw ing ▁as ▁in ▁the ▁lower ▁image , ▁we ▁see ▁that ▁this ▁figure ▁consists ▁of ▁two ▁squ ares , ▁one ▁of ▁area ▁a ^{2} ▁and ▁the ▁other ▁of ▁area ▁b ^{2} .
▁Then ▁C ind y ▁dou bles ▁the ▁number ▁of ▁pens ▁you ▁have , ▁so ▁you ▁have ▁2 ▁* ▁(20 ▁+ ▁M ) ▁pens .
▁Total ▁cost ▁before ▁transportation ▁= ▁P urch ase ▁price ▁+ ▁Rep air ▁cost
▁1. ▁If ▁you ▁move ▁a ▁shape ▁rigid ly ▁( with out ▁stret ching ▁or ▁dist ort ing ▁it ), ▁then ▁its ▁area ▁does ▁not ▁change .
▁V 2 ▁in ▁km / h ▁= ▁80 .01 ▁km / h ▁( app ro xim ately )
▁Let ▁us ▁think ▁a ▁little ▁more ▁on ▁what ▁is ▁meant ▁by ▁being ▁meas ur able , ▁taking ▁lengths ▁as ▁an ▁example .
▁Note ▁that ▁if ▁r = 1, ▁nothing ▁changes ; ▁this ▁" " d ilation " ▁is ▁called ▁the ▁ident ity .
▁The ▁number ▁of ▁smaller ▁pieces ▁is ▁2 ^ n , ▁where ▁n ▁is ▁the ▁number ▁of ▁f old s .
▁This ▁fundamental ▁ident ity ▁is ▁one ▁of ▁the ▁reasons ▁why ▁so ▁many ▁geomet r ical ▁truth s ▁can ▁be ▁put ▁into ▁an ▁algebraic ▁dress .
▁Since ▁2 .25 x ▁40 = 9 0, ▁they ▁will ▁need ▁90 ▁bags ▁of ▁sand ▁for ▁the ▁large ▁sw ing ▁set .
▁Let ' s ▁first ▁determine ▁the ▁rati os ▁for ▁the ▁standard ▁form ulation ▁and ▁then ▁adjust ▁them ▁for ▁the ▁s port ▁form ulation .
▁The ▁f rog ▁j ump ed ▁0.9 5 ▁meters , ▁so ▁in ▁inches ▁that ▁would ▁be :
▁First , ▁calculate ▁the ▁difference ▁between ▁the ▁final ▁and ▁initial ▁values :
▁In ▁real ▁life ▁problems , ▁the ▁unit ▁square ▁could ▁be ▁one ▁in ch , ▁one ▁centim eter , ▁one ▁yard , ▁or ▁even ▁one ▁m ile ▁on ▁each ▁side , ▁depending ▁on ▁what ▁you ▁are ▁meas uring .
▁On ▁the ▁second ▁day , ▁1000 ▁people ▁took ▁2 ▁can s ▁each , ▁which ▁means :
▁Se lling ▁Pr ice ▁of ▁the ▁second ▁car ▁( SP 2) ▁= ▁C P 2 ▁- ▁(1 3 % ▁of ▁C P 2)
▁Each ▁can ▁of ▁yellow ▁paint ▁co vers ▁1.5 ▁square ▁meters , ▁so ▁the ▁number ▁of ▁can s ▁needed ▁for ▁yellow ▁paint ▁is :
▁Bet ty ▁e ats ▁3 ▁cookies ▁a ▁day , ▁so ▁in ▁a ▁week ▁(7 ▁days ), ▁she ▁will ▁eat ▁3 ▁cookies / day ▁* ▁7 ▁days ▁= ▁21 ▁cookies .
▁First , ▁let ' s ▁convert ▁all ▁the ▁given ▁amount s ▁to ▁o un ces .
▁Now ▁move ▁the ▁smaller ▁triangle ▁by ▁a ▁rot ation ▁with ▁center ▁A = A ', ▁so ▁that ▁the ▁point ▁C ▁lands ▁on ▁the ▁segment ▁A ' C '.
▁- ▁B ars ▁taken ▁by ▁Thomas ▁and ▁friends ▁= ▁300
▁The ▁coordinate ▁rule ▁for ▁the ▁reflection ▁in ▁the ▁x - axis ▁is ▁( x , y )- > ( x , - y ).
▁If ▁5 ▁hours ▁is ▁to ▁N /4 ▁nut s , ▁then ▁4 ▁hours ▁is ▁to ▁X ▁nut s ▁( where ▁X ▁is ▁the ▁number ▁of ▁nut s ▁the ▁c row ▁e ats ▁in ▁4 ▁hours ).
▁Spe ed ▁of ▁Pol ly ▁= ▁Distance ▁/ ▁Time ▁= ▁3 ▁miles ▁/ ▁0.5 ▁hours ▁= ▁6 ▁miles ▁per ▁hour
▁In ▁both ▁cases , ▁just ▁measure ▁dist ances ▁along ▁horizontal ▁and ▁vertical ▁l ens ▁and ▁use ▁the ▁Py th ag ore an ▁the ore m .
▁The ▁cub e ▁root ▁of ▁a ▁common ▁fraction ▁whose ▁denominator ▁is ▁not ▁a ▁perfect ▁cub e ▁can ▁be ▁found ▁approximately ▁by ▁reducing ▁the ▁fraction ▁to ▁a ▁decimal , ▁and ▁then ▁ext ract ing ▁the ▁root .
▁Finally , ▁we ▁can ▁divide ▁this ▁by ▁15 ▁to ▁find ▁the ▁mean ▁profit ▁for ▁the ▁last ▁15 ▁days .
▁Therefore , ▁the ▁friend ▁sold ▁the ▁car ▁for ▁Rs . ▁54 000 .
▁Do ing ▁the ▁multip lication ▁we ▁have ▁two ▁faces ▁of ▁88 ▁s q .
▁G er ald ' s ▁average ▁speed ▁was ▁given ▁as ▁3 ▁miles ▁per ▁hour .
▁Although ▁Mat hem at ics ▁te nds ▁to ▁be ▁quite ▁rig orous ▁in ▁the ▁construction ▁of ▁formal ▁pro of s , ▁we ▁know ▁through ▁experience , ▁that ▁inform al , ▁int u itive ▁understanding ▁of ▁the ▁" wh y " ▁of ▁a ▁pro of ▁always ▁preced es ▁its ▁art ic ulation .
▁Total ▁turn s ▁= ▁Total ▁capacity ▁of ▁the ▁dr um ▁/ ▁Com b ined ▁capacity ▁per ▁turn
▁- ▁Employ ed ▁mal es ▁= ▁Total ▁employ ed ▁- ▁Employ ed ▁fem ales
▁The ▁three - point ▁line ▁is ▁the ▁line ▁that ▁separ ates ▁the ▁two - point ▁area ▁from ▁the ▁three - point ▁area ; ▁any ▁shot ▁converted ▁beyond ▁this ▁line ▁count s ▁as ▁three ▁points .
▁The ▁1 x ▁1 x ▁24 ▁pr ism ▁has ▁22 ▁cub es ▁with ▁four ▁exp osed ▁faces ▁and ▁the ▁two ▁end ▁cub es ▁have ▁five ▁exp osed ▁faces .
▁These ▁are ▁the ▁col oring ▁books ▁left ▁after ▁the ▁store ▁got ▁rid ▁of ▁thirty - three ▁of ▁them .
▁An ▁equ ilater al ▁triangle ▁can ▁be ▁obtained ▁by ▁a ▁plane ▁section ▁by ▁cutting ▁the ▁cub e ▁by ▁a ▁plane ▁that ▁is ▁per pend ic ular ▁to ▁the ▁dia gon al ▁jo ining ▁two ▁opp osed ▁vert ices ▁of ▁the ▁cub e .
▁From ▁the ▁above ▁two ▁equations , ▁we ▁can ▁find ▁the ▁work ▁rate ▁of ▁b ▁by ▁subtract ing ▁the ▁work ▁rate ▁of ▁a ▁from ▁the ▁combined ▁work ▁rate ▁of ▁a ▁and ▁b :
▁D il ations ▁are ▁also ▁directly ▁connected ▁to ▁scale ▁changes .
▁The ▁total ▁time ▁taken ▁to ▁complete ▁the ▁work ▁is ▁75 ▁days , ▁and ▁since ▁Am it ▁worked ▁for ▁3 ▁days , ▁An an th u ▁worked ▁for :
▁What ▁is ▁the ▁nature ▁of ▁the ▁cor rel ation ▁between ▁the ▁triangles , ▁when ▁the ▁three ▁angles ▁of ▁the ▁one ▁are ▁respectively ▁equal ▁to ▁the ▁three ▁angles ▁of ▁the ▁other ?
▁2. ▁J imin ▁has ▁more ▁apples ▁than ▁Y o ong i .
▁Now , ▁refer ring ▁back ▁to ▁Figure ▁2, ▁when ▁we ▁drop ▁the ▁per pend ic ular ▁we ▁divide ▁the ▁triangle ▁into ▁two ▁right ▁triangles , ▁the ▁sum ▁( or ▁difference , ▁depending ▁upon ▁whether ▁the ▁original ▁triangle ▁is ▁acute ▁or ▁ob t use ) ▁of ▁whose ▁b ases ▁is ▁b , ▁so ▁the ▁general ▁fact ▁hold s .
▁The ▁resulting ▁3 D ▁figure ▁is ▁called ▁a ▁py ram id .
▁The ▁rate ▁is ▁the ▁fraction ▁of ▁work ▁done ▁per ▁day , ▁which ▁is ▁(2 / 3) ▁/ ▁12 ▁days .
▁Therefore , ▁B ▁ear ns ▁Rs . ▁100 ▁more ▁than ▁A .
▁Total ▁Y ellow ▁P aint ▁Are a ▁= ▁W all ▁C ▁+ ▁W all ▁D ▁+ ▁W all ▁E ▁= ▁8 .75 m 2 ▁+ ▁5 .5 m 2 ▁+ ▁12.5 m 2 ▁= ▁26 .75 m 2
▁The ▁E uc l ide an ▁pro of ▁of ▁this ▁statement ▁is ▁an ▁application ▁of ▁S S S ▁for ▁triangles .
▁8 000 ▁gallons ▁needed ▁/ ▁400 ▁gallons ▁produced ▁per ▁day ▁= ▁20 ▁days
▁Now ▁we ▁have ▁the ▁s port ▁form ulation ▁rati os ▁as ▁follows :
▁In ▁this ▁case , ▁the ▁number ▁of ▁terms ▁is ▁16 , ▁the ▁first ▁term ▁is ▁" n ", ▁and ▁the ▁last ▁term ▁is ▁" n ▁+ ▁30 ".
▁so ▁we ▁must ▁have ▁B =10 8 ▁s q .
▁4 ▁acres / day ▁* ▁400 ▁gallons / ac re ▁= ▁16 00 ▁gallons / day
▁P ick ▁a ▁ver te x , ▁let ' s ▁say ▁A , ▁and ▁consider ▁the ▁three ▁ed ges ▁meeting ▁at ▁the ▁ver te x .
▁The ▁ratio ▁of ▁areas ▁is
▁F ind ing ▁and ▁demonst r ating ▁criteria ▁for ▁the ▁con g ru ence ▁of ▁triangles ▁is ▁a ▁major ▁top ic ▁in ▁Second ary ▁I ; ▁as ▁an ▁illust ration , ▁here ▁we ▁show ▁how ▁one ▁such ▁criter ion , ▁known ▁as ▁S S S , ▁follows ▁from ▁the ▁properties ▁of ▁rigid ▁mot ions .
▁2. ▁Ge omet ric ▁Se qu ence : ▁Each ▁number ▁is ▁a ▁constant ▁multiple ▁of ▁the ▁previous ▁number .
▁(1 1 ▁pregnant ▁female ▁c ows ▁/ ▁22 ▁female ▁c ows ) ▁* ▁100 ▁= ▁50%
▁First ▁of ▁all , ▁there ▁is ▁the ▁translation ▁of ▁P ▁to ▁Q , ▁and ▁we ▁can ▁follow ▁that ▁by ▁any ▁rot ation ▁about ▁the ▁point ▁Q .
▁The ▁at omic ▁weight ▁of ▁79 .9 6 ▁am u ▁is ▁cl os est ▁to ▁the ▁at omic ▁weight ▁of ▁b rom ine ▁( B r ), ▁which ▁is ▁approximately ▁79 .9 0 4 ▁am u .
▁The ▁answer ▁is ▁yes : ▁the ▁difference ▁between ▁Figure ▁6 a ▁and ▁that ▁of ▁Figure ▁7 ▁is ▁that ▁of ▁o ri ent ation .
▁After ▁adding ▁the ▁2 ▁kil ogram s ▁of ▁solution ▁Y , ▁the ▁new ▁solution ▁will ▁have :
▁This ▁is ▁what ▁is ▁called ▁the ▁con verse ▁of ▁the ▁Py th ag ore an ▁the ore m .
▁This ▁means ▁that ▁no ▁men ▁should ▁leave ▁the ▁work ▁after ▁14 ▁days , ▁as ▁the ▁original ▁number ▁of ▁men ▁( 30) ▁is ▁exactly ▁what ▁is ▁needed ▁to ▁complete ▁the ▁work ▁in ▁the ▁remaining ▁time .
▁So ▁the ▁original ▁triangles ▁are ▁con g ru ent .
▁Since ▁Peter ▁and ▁Al ice ▁get ▁3 ▁extra ▁b ars ▁each , ▁they ▁will ▁take ▁the ▁remainder :
▁First ▁of ▁all , ▁triangle ▁I ▁is ▁con g ru ent ▁to ▁triangle ▁II : ▁transl ate ▁triangle ▁I ▁hor izon t ally ▁so ▁that ▁the ▁side ▁lab el ed ▁" a " ▁lies ▁on ▁the ▁side ▁of ▁triangle ▁II ▁lab el ed ▁" b .
▁She ▁starts ▁with ▁60 ▁cookies ▁and ▁after ▁eating ▁21, ▁she ▁will ▁have ▁60 ▁cookies ▁- ▁21 ▁cookies ▁= ▁39 ▁cookies ▁left .
▁The ▁difference ▁between ▁the ▁number ▁of ▁vert ices ▁of ▁a ▁he x ag on al ▁pr ism ▁and ▁a ▁p ent ag on al ▁pr ism ▁is ▁12 ▁- ▁10 ▁= ▁2 ▁vert ices .
▁The ▁rest ▁of ▁the ▁big ▁square ▁of ▁Figure ▁9 ▁consists ▁of ▁four ▁triangles , ▁so ▁what ▁is ▁outside ▁those ▁triangles ▁has ▁area ▁a ^{2} + b ^{2} .
▁Since ▁one ▁not ep ad ▁contains ▁40 ▁notes , ▁this ▁means ▁that ▁the ▁5 ▁pieces ▁of ▁paper ▁must ▁y ield ▁40 ▁smaller ▁note ▁papers ▁once ▁f olded ▁and ▁cut .
▁To ▁be ▁a ▁square , ▁we ▁have ▁to ▁show ▁that ▁one ▁of ▁the ▁corn er ▁angles ▁is ▁a ▁right ▁angle .
▁What ▁is ▁the ▁volume ▁of ▁the ▁box , ▁and ▁what ▁is ▁its ▁surface ▁area ?
▁Since ▁64 ▁is ▁greater ▁than ▁6, ▁the ▁smallest ▁integer ▁k ▁for ▁which ▁64 ^ k ▁> ▁6 ▁is ▁k ▁= ▁1.
▁Note ▁that , ▁if ▁the ▁d ilation ▁that ▁places ▁T ▁on to ▁T ' ▁has ▁factor ▁r , ▁then ▁the ▁factor ▁of ▁the ▁d ilation ▁pl ac ing ▁T ' ▁on to ▁T ▁is ▁r ^{ -1 } .
▁V er ify ▁by ▁measurement ▁that ▁triangle ▁ABC ▁is ▁a ▁right ▁triangle .
▁We ▁have ▁5 ▁liters ▁of ▁the ▁first ▁solution ▁with ▁x % ▁silver ▁and ▁we ▁need ▁to ▁add ▁2.5 ▁liters ▁of ▁a ▁10% ▁silver ▁solution ▁to ▁get ▁a ▁final ▁solution ▁that ▁is ▁6% ▁silver .
▁Another ▁triangle ▁has ▁leg ▁lengths ▁20 ▁in ▁and ▁25 ▁in .
▁3 ▁slic es / we ek end ▁* ▁52 ▁week end s / year ▁= ▁15 6 ▁slic es / year
▁To ▁find ▁the ▁mole c ular ▁weight ▁of ▁car bon ic ▁ac id ▁( H 2 CO 3 ), ▁we ▁can ▁use ▁the ▁information ▁given ▁for ▁8 ▁m oles .
▁Jo an ▁sold ▁26 ▁books ▁in ▁the ▁yard ▁sale .
▁Let ' s ▁try ▁the ▁basic ▁operations :
▁So , ▁the ▁number ▁of ▁rot ten ▁apples ▁is ▁0.4 0 A .
▁I ▁was ▁planning ▁to ▁send ▁a ▁g ad get ▁that ▁is ▁8 ▁in .
▁Since ▁we ▁want ▁to ▁find ▁the ▁length ▁of ▁C D , ▁let ' s ▁denote ▁that ▁number ▁by ▁h .
▁From ▁the ▁smaller ▁po nd , ▁we ▁know ▁that ▁20% ▁of ▁the ▁30 ▁d uc ks ▁are ▁green , ▁which ▁means :
▁Se an ▁played ▁for ▁700 ▁minutes ▁and ▁Ind ira ▁played ▁for ▁812 ▁minutes , ▁so ▁together ▁they ▁played :
▁4 96 ▁grams ▁/ ▁8 ▁m oles ▁= ▁62 ▁grams / m ole
▁We ▁know ▁the ▁number ▁of ▁green ▁d uc ks ▁from ▁both ▁po nds ▁is ▁the ▁sum ▁of ▁green ▁d uc ks ▁from ▁the ▁smaller ▁po nd ▁and ▁green ▁d uc ks ▁from ▁the ▁larger ▁po nd , ▁so :
▁- ▁P ages ▁1 ▁through ▁9 ▁require ▁1 ▁dig it ▁each .
▁For ▁the ▁prime ▁factor ▁3, ▁which ▁has ▁an ▁exp on ent ▁of ▁3, ▁we ▁have ▁3 ▁+ ▁1 ▁= ▁4 ▁cho ices ▁( inc lud ing ▁not ▁cho osing ▁it ▁at ▁all ).
▁N ed ▁ordered ▁11 ▁p iz z as ▁for ▁the ▁party .
▁30 ▁stick ers ▁( red ) ▁+ ▁20 ▁stick ers ▁( g re en ) ▁+ ▁10 ▁stick ers ▁( b l ue ) ▁= ▁60 ▁stick ers
▁Figure ▁11 ▁depic ts ▁a ▁typ ical ▁poly hed ron .
▁Now , ▁let ' s ▁add ▁the ▁stick ers ▁from ▁all ▁the ▁f old ers ▁to ▁find ▁the ▁total ▁number ▁of ▁stick ers ▁Pe gg y ▁us es :
▁Therefore , ▁there ▁were ▁200 ▁apples ▁in ▁the ▁or ch ard .
▁T aking ▁square ▁roots , ▁we ▁have ▁h =2 25 ▁approximately .
▁He ▁sold ▁29 ▁of ▁them .
▁Let ' s ▁denote ▁the ▁weight ▁T ony ▁can ▁lif t ▁in ▁the ▁cur l ▁exerc ise ▁as ▁C ▁p ounds .
▁Since ▁13 ▁lo aves ▁would ▁only ▁provide ▁15 6 ▁slic es ▁and ▁she ▁needs ▁15 6, ▁she ▁will ▁need ▁to ▁purch ase ▁13 ▁lo aves ▁exactly , ▁as ▁it ▁div ides ▁even ly ▁with ▁no ▁remainder .
▁Total ▁work ▁= ▁1200 ▁work ▁units
▁First , ▁let ' s ▁find ▁out ▁how ▁many ▁stra w ber ries ▁and ▁or anges ▁Kar le e ▁has .
▁After ▁80 ▁new ▁board ers ▁join ▁the ▁school , ▁the ▁number ▁of ▁board ers ▁becomes ▁B ▁+ ▁80 .
▁9 ▁games / mon th ▁* ▁3 ▁months ▁= ▁27 ▁games
▁Therefore , ▁the ▁number ▁that ▁is ▁15% ▁of ▁80 ▁is ▁12.
▁Thus ▁V ▁is ▁the ▁square ▁root ▁of ▁the ▁pro d ut ▁of ▁the ▁sides .
▁What ▁does ▁it ▁mean ▁to ▁say ▁that ▁the ▁area ▁of ▁a ▁region ▁is ▁18 ▁square ▁inches ?
▁Let ' s ▁break ▁down ▁K im ' s ▁morning ▁routine ▁into ▁individual ▁t as ks ▁and ▁calculate ▁the ▁time ▁spent ▁on ▁each :
▁So ▁the ▁image ▁of ▁C ▁under ▁that ▁d ilation ▁lies ▁on ▁the ▁line ▁of ▁B ' C ' ▁and ▁the ▁line ▁of ▁AC ', ▁so ▁it ▁has ▁to ▁be ▁C '.
▁Since ▁the ▁top ▁has ▁the ▁same ▁area ▁as ▁the ▁bott om , ▁and ▁there ▁are ▁4 ▁sides , ▁the ▁surface ▁area ▁is ▁A ▁= ▁2 ( 100 ) ▁+ ▁4 ( 60 ) ▁= ▁4 40 ▁s q .
▁But ▁now ▁we ▁can ▁see ▁that ▁the ▁d ilation ▁with ▁center ▁A ▁that ▁mo ves ▁B ▁to ▁B ' ▁put s ▁the ▁smaller ▁triangle ▁on ▁top ▁of ▁the ▁larger ▁one .
▁To ▁find ▁12 % ▁of ▁x , ▁we ▁convert ▁12 % ▁to ▁a ▁decimal :
▁Inte rest ▁= ▁Face ▁Val ue ▁* ▁Inte rest ▁R ate
▁This ▁is ▁an ▁arithmetic ▁series ▁with ▁16 ▁terms .
▁5 ▁pot at oes ▁( l unch ) ▁+ ▁2 ▁pot at oes ▁( d in ner ) ▁= ▁7 ▁pot at oes ▁total .
▁- ▁Each ▁successful ▁extra ▁point ▁is ▁worth ▁1 ▁point , ▁so ▁2 ▁successful ▁extra ▁points ▁would ▁be ▁2 ▁* ▁1 ▁= ▁2 ▁points .
▁P it ts b ur gh ▁to ▁Prov idence : ▁Four ▁and ▁a ▁quarter ▁inches , ▁or ▁4 25 ▁miles .
▁We ▁have ▁indicated ▁the ▁correspond ing ▁angles ▁with ▁the ▁g reek ▁letters ▁al ph a , b et a , g am ma .
▁Thus ▁the ▁scale ▁of ▁this ▁phot o ▁( at ▁the ▁vol c an o ) ▁is ▁12 ▁mm ▁: ▁6 000 ▁feet , ▁or ▁1 ▁mm ▁: ▁500 ▁feet .
▁Cash ▁D isc ount ▁= ▁L isted ▁Pr ice ▁- ▁E ffect ive ▁Se lling ▁Pr ice
▁As ▁it ▁turn s ▁out , ▁that ▁g ad get ▁is ▁no ▁longer ▁available , ▁but ▁I ▁can ▁send ▁her ▁a ▁scale - red u ced ▁model ▁of ▁the ▁g ad get , ▁reduced ▁to ▁60 % ▁in ▁every ▁linear ▁d im ension .
▁Then , ▁the ▁numbers ▁we ▁are ▁looking ▁for ▁are ▁of ▁the ▁form ▁2 ▁* ▁x ▁* ▁n , ▁where ▁n ▁is ▁a ▁positive ▁integer .
▁F ind ▁the ▁area ▁of ▁each ▁face ▁and ▁add :
▁- ▁Per cent ▁of ▁employ ed ▁mal es ▁= ▁( E mp loy ed ▁mal es ▁/ ▁Total ▁population ) ▁* ▁100
▁Total ▁b ars ▁taken ▁by ▁Thomas ▁and ▁friends ▁= ▁3 / 8 ▁* ▁800 ▁= ▁300 ▁b ars
▁Using ▁a ▁ruler , ▁we ▁find ▁that ▁the ▁height ▁of ▁the ▁image ▁of ▁the ▁vol c an o ▁is ▁12 ▁mm , ▁the ▁length ▁of ▁the ▁left ▁slope ▁is ▁21 ▁mm , ▁and ▁the ▁pl ume ▁is ▁10 ▁mm ▁high .
▁This ▁gives ▁us ▁the ▁full ▁lab ell ing ▁of ▁Figure ▁11.
▁According ▁to ▁the ▁given ▁information , ▁8 ▁years ▁ago , ▁the ▁sum ▁of ▁their ▁ages ▁was ▁7 6.
▁R ac hel ▁bought ▁a ▁total ▁of ▁7 ▁ch airs ▁+ ▁3 ▁tables ▁= ▁10 ▁pieces ▁of ▁furn iture .
▁21 ▁video ▁f iles ▁* ▁50 ▁M B ▁each ▁= ▁10 50 ▁M B
▁Now , ▁Figure ▁17 ▁is ▁of ▁a ▁pr ism ▁over ▁a ▁tra pe z oid al ▁base ▁B .
▁The ▁total ▁number ▁of ▁b ul bs ▁that ▁are ▁sw it ched ▁on ▁is ▁70% ▁of ▁all ▁the ▁b ul bs , ▁which ▁is ▁0.7 0 ( I ▁+ ▁F ).
▁Therefore , ▁the ▁new ▁solution ▁is ▁36 % ▁liqu id ▁X .
▁When ▁the ▁park ▁workers ▁fin ish ▁plant ing ▁the ▁trees , ▁the ▁park ▁will ▁have :
▁So , ▁Al issa ▁needs ▁to ▁f old ▁the ▁paper ▁3 ▁times ▁to ▁get ▁the ▁required ▁40 ▁notes ▁from ▁5 ▁pieces ▁of ▁letter - s ize ▁paper .
▁However , ▁since ▁you ▁can ' t ▁purch ase ▁a ▁fraction ▁of ▁a ▁lo af , ▁Su z an ne ▁will ▁need ▁to ▁purch ase ▁whole ▁lo aves .
▁Let s ▁concent rate ▁on ▁the ▁ver te x ▁marked ▁A , ▁where ▁the ▁angle ▁is ▁marked ▁g am ma .
▁When ▁we ▁think ▁of ▁“ two ” ▁and ▁“ three ” ▁we ▁see ▁stro k es ▁in ▁a ▁row , ▁or ▁balls ▁in ▁a ▁he ap , ▁or ▁some ▁other ▁physical ▁ag g reg ation ▁of ▁particular ▁things .
▁The ▁length ▁whose ▁measure ▁we ▁want ▁is ▁the ▁hypot en use ▁of ▁a ▁triangle ▁D elt a ▁AC B ▁whose ▁leg ▁lengths ▁are ▁23 .3 ▁and ▁8 ▁feet .
▁Let ' s ▁list ▁the ▁equivalent ▁fractions ▁with ▁two - d ig it ▁denomin ators :
▁The ▁second ▁has ▁the ▁advant age ▁that ▁it ▁can ▁be ▁general ized ▁to ▁poly g ons ▁with ▁more ▁sides .
▁More ▁spec if ically , ▁it ▁takes ▁segment ▁V A ▁to ▁V C ▁and ▁V B ▁to ▁V D , ▁and ▁so ▁car ries ▁AVB ▁to ▁C VD , .
▁So ▁the ▁length ▁of ▁P R ▁is ▁4, ▁and ▁the ▁length ▁of ▁Q R ▁is ▁3.
▁If ▁John ▁has ▁to ▁watch ▁2 ▁hours ▁a ▁day ▁for ▁5 ▁days , ▁then ▁the ▁total ▁time ▁he ▁will ▁spend ▁watch ing ▁the ▁show ▁is :
▁For ▁n ▁= ▁2: ▁5 ▁* ▁2 ^2 ▁= ▁5 ▁* ▁4 ▁= ▁20 ▁( st ill ▁not ▁enough )
▁According ▁to ▁the ▁information ▁given , ▁40% ▁of ▁the ▁apples ▁were ▁rot ten .
▁This ▁approxim ate ▁values ▁are ▁close ▁enough ▁to ▁confirm ▁the ▁distance ▁calculation , ▁and ▁attrib ute ▁the ▁disc rep ancy ▁to ▁minor ▁imprecision ▁in ▁measurement .
▁3 ▁m oles ▁C 6 H 5 C l ▁× ▁1 ▁mole ▁Cl 2/ m ole ▁C 6 H 5 C l ▁= ▁3 ▁m oles ▁Cl 2
▁If ▁a ▁boy ▁receiv es ▁2 ▁c ents ▁for ▁his ▁first ▁day ’ s ▁work , ▁4 ▁c ents ▁for ▁his ▁second ▁day , ▁8 ▁c ents ▁for ▁the ▁third ▁day , ▁and ▁so ▁on ▁for ▁12 ▁days , ▁what ▁will ▁his ▁w ages ▁amount ▁to ?
▁Total ▁blue ▁ball oons ▁= ▁A ly ss a ' s ▁blue ▁+ ▁S andy ' s ▁blue ▁+ ▁S ally ' s ▁blue
▁In gr id ▁arrived ▁at ▁10 : 45 ▁a . m ., ▁so ▁D il an ▁arrived ▁at ▁10 : 35 ▁a . m .
▁Now ▁we ▁know ▁that ▁Car ▁B ▁co vers ▁a ▁distance ▁of ▁100 ▁km ▁in ▁4 ▁hours .
▁Therefore , ▁the ▁van ▁should ▁maintain ▁a ▁speed ▁of ▁20 ▁km / h ▁to ▁cover ▁the ▁same ▁distance ▁in ▁9 ▁hours .
▁For ▁any ▁two ▁points ▁P ▁and ▁Q , ▁the ▁distance ▁between ▁P ▁and ▁Q ▁is ▁the ▁length ▁of ▁the ▁line ▁segment ▁P Q .
▁We ▁conclud e ▁that ▁for ▁any ▁triangle ▁with ▁side ▁lengths ▁a ▁and ▁b , ▁the ▁length ▁c ▁of ▁the ▁third ▁side ▁is ▁either ▁less ▁than ▁s q r t { a ^{2} + b ^{2} } ▁( t ri angle ▁is ▁acute ), ▁or ▁greater ▁than ▁s q r t { a ^{2} + b ^{2} } ▁( t ri angle ▁is ▁ob t use ) ▁except ▁for ▁the ▁right ▁triangle ( where ▁the ▁segment ▁AB ▁is ▁vertical ).
▁- ▁For ▁page ▁100 000 , ▁there ▁is ▁1 ▁page ▁requ iring ▁6 ▁dig its : ▁1 ▁* ▁6 ▁= ▁6 ▁dig its
▁- ▁The ▁third ▁number ▁is ▁14 .
▁Total ▁b ars ▁accounted ▁for ▁without ▁extra ▁= ▁33 ▁* ▁9 ▁= ▁29 7 ▁b ars
▁What ▁is ▁the ▁length ▁of ▁the ▁alt itude ▁( CD ) ▁of ▁the ▁triangle ?
▁Now ▁draw ▁the ▁horizontal ▁line ▁starting ▁at ▁P ▁and ▁the ▁vertical ▁line ▁starting ▁at ▁Q ▁and ▁let ▁R ▁be ▁the ▁point ▁of ▁inter section .
▁central ▁rectangle : ▁area ▁is ▁3 ▁s q .
▁53 ▁( total ▁students ) ▁- ▁4 ▁( st ud ents ▁a head ▁of ▁her ) ▁+ ▁1 ▁( J aya ▁herself ) ▁= ▁50
▁These ▁ex amples ▁show ▁us ▁that ▁the ▁distance ▁between ▁two ▁points ▁in ▁the ▁plane ▁can ▁be ▁calculated ▁using ▁the ▁Py th ag ore an ▁the ore m , ▁since ▁the ▁slope ▁triangle ▁with ▁hypot en use ▁the ▁line ▁segment ▁jo ining ▁the ▁two ▁points ▁is ▁a ▁right ▁angle .
▁Both ▁addition ▁and ▁multip lication ▁seem ▁to ▁work , ▁but ▁since ▁we ▁are ▁looking ▁for ▁a ▁single ▁operation , ▁we ▁should ▁choose ▁the ▁one ▁that ▁f its ▁the ▁context ▁of ▁the ▁problem .
▁The ▁same ▁method ▁will ▁apply ▁to ▁numbers ▁of ▁more ▁than ▁two ▁groups ▁of ▁figures ▁by ▁considering ▁a ▁in ▁the ▁typ ical ▁form ▁to ▁represent ▁at ▁each ▁step ▁the ▁part ▁of ▁the ▁root ▁already ▁found .
▁She ▁starts ▁with ▁10 ▁bro wn ies ▁and ▁after ▁eating ▁7, ▁she ▁will ▁have ▁10 ▁bro wn ies ▁- ▁7 ▁bro wn ies ▁= ▁3 ▁bro wn ies ▁left .
▁To ▁find ▁B ren ' s ▁age ▁10 ▁years ▁from ▁now , ▁we ▁simply ▁add ▁10 ▁to ▁his ▁current ▁age :
▁A ▁corn er ▁point ▁where ▁several ▁faces ▁come ▁together ▁is ▁called ▁a ▁ver te x ▁( vert ices ▁is ▁the ▁pl ural ) ▁or ▁corn er ▁of ▁the ▁poly hed ron .
▁Now , ▁the ▁friend ▁sell s ▁the ▁car ▁at ▁a ▁20% ▁gain .
▁The ▁entire ▁court ▁is ▁50 ▁feet ▁by ▁94 ▁feet .
▁Two ▁angles ▁are ▁con g ru ent ▁if ▁they ▁have ▁the ▁same ▁measure .
▁C ody ▁origin ally ▁had ▁18 ▁marbles .
▁This ▁can ▁be ▁shown ▁by ▁creating ▁cont ain ers ▁with ▁these ▁sh ap es ▁and ▁comparing ▁the ▁volume ▁of ▁water ▁that ▁f ills ▁the ▁objects .
▁O ak ▁trees : ▁7 ▁( c urrent ) ▁+ ▁6 ▁( pl anted ) ▁= ▁13 ▁o ak ▁trees
▁Let ' s ▁use ▁the ▁downstream ▁equation :
▁A ▁right ▁triangle ▁has ▁leg ▁lengths ▁6 ▁in ▁and ▁8 ▁in .
▁Then ▁we ▁find ▁the ▁area ▁of ▁each ▁shape .
▁So , ▁the ▁park ▁will ▁have ▁11 ▁m ap le ▁trees , ▁17 ▁pop l ar ▁trees , ▁and ▁13 ▁o ak ▁trees ▁when ▁the ▁workers ▁are ▁finished .
▁1. ▁Ar ithmetic ▁Se qu ence : ▁The ▁numbers ▁increase ▁or ▁decrease ▁by ▁a ▁constant ▁difference .
▁If ▁And y ▁can ▁get ▁50 ▁dem er its ▁before ▁getting ▁f ired , ▁and ▁his ▁allow able ▁dem er its ▁are ▁reduced ▁by ▁10 ▁due ▁to ▁the ▁performance ▁evaluation , ▁his ▁new ▁limit ▁is :
▁However , ▁this ▁count s ▁each ▁game ▁twice ▁( on ce ▁for ▁each ▁team ▁involved ), ▁so ▁we ▁need ▁to ▁divide ▁by ▁2 ▁to ▁get ▁the ▁actual ▁number ▁of ▁games :
▁If ▁P ra k ash ▁can ▁copy ▁18 ▁pages ▁in ▁2 ▁hours , ▁then ▁in ▁1 ▁hour ▁he ▁can ▁copy :
▁C hem other ap y ▁treatment ▁of ▁cancer ▁takes ▁place ▁through ▁the ▁surface ▁of ▁the ▁growth , ▁so ▁the ▁surface ▁area ▁of ▁the ▁cancer ▁is ▁a ▁more ▁important ▁par am eter ▁than ▁its ▁volume .
▁The ▁figure ▁above ▁is ▁a ▁gener ic ▁py ram id ▁whose ▁base ▁is ▁a ▁poly gon ▁with ▁five ▁sides .
▁We ▁are ▁told ▁that ▁T ony ▁can ▁lif t ▁900 ▁p ounds ▁in ▁the ▁squ at ▁exerc ise .
▁Prof it ▁or ▁L oss ▁Per cent ▁= ▁( Pro f it ▁or ▁L oss ▁/ ▁T C P ) ▁* ▁100
▁4 ▁music ▁f iles ▁* ▁5 ▁M B ▁each ▁= ▁20 ▁M B
▁First , ▁transl ate ▁A ▁to ▁A '.
▁Cost ▁per ▁b age l ▁in ▁a ▁doz en ▁= ▁ € 38 ▁/ ▁12
▁Since ▁we ▁don ' t ▁know ▁the ▁common ▁difference , ▁we ▁can ' t ▁determine ▁the ▁exact ▁numbers ▁without ▁more ▁information .
▁a ▁triangle ▁that ▁is ▁not ▁equ ilater al
▁Supp ose ▁we ▁double ▁the ▁lengths ▁of ▁the ▁legs ▁of ▁a ▁right ▁triangle .
▁Finally , ▁let ' s ▁calculate ▁the ▁combined ▁total ▁score ▁for ▁both ▁teams : ▁32 ▁( H aw ks ) ▁+ ▁30 ▁( E ag les ) ▁= ▁62 ▁points .
▁F ind ▁the ▁sum ▁of ▁8 ▁terms ▁of ▁the ▁series ▁14 .
▁What ▁is ▁the ▁volume ▁of ▁gas ▁that ▁was ▁used ▁to ▁fully ▁infl ate ▁the ▁f ake ▁Py ram id ?
▁Total ▁pages ▁= ▁(2 5 /3 ▁pages ▁per ▁minute ) ▁* ▁6 ▁minutes ▁= ▁50 ▁pages
▁So , ▁50 ▁percent ▁of ▁the ▁female ▁c ows ▁are ▁pregnant .
▁4 ▁m oles ▁CH 3 B r ▁× ▁1 ▁mole ▁Br 2/ m ole ▁CH 3 B r ▁= ▁4 ▁m oles ▁Br 2
▁Now ▁rot ation ▁with ▁ver te x ▁V ▁through ▁a ▁straight ▁angle ▁(1 80 degrees ) ▁takes ▁the ▁line ▁AC ▁into ▁itself .
▁Now , ▁the ▁d ash - d ot ▁line ▁on ▁the ▁floor ▁of ▁the ▁room ▁is ▁the ▁hypot en use ▁of ▁a ▁right ▁triangle ▁of ▁leg ▁lengths ▁12 ▁f t ▁and ▁20 ▁f t .
▁The ▁cost ▁per ▁b age l ▁when ▁buy ing ▁ind iv id ually ▁is ▁ € 3 .50 .
▁Work ▁done ▁by ▁Am it ▁+ ▁Work ▁done ▁by ▁An an th u ▁= ▁W
▁How ▁much ▁pack aging ▁material ▁was ▁used ▁to ▁constru ct ▁the ▁package
▁p iz za ▁has ▁the ▁same ▁th ick ness ▁as ▁a ▁10 ▁in .
▁The ▁least ▁multiple ▁that ▁satisf ies ▁this ▁condition ▁is ▁the ▁product ▁of ▁these ▁prime ▁factors ▁themselves :
▁S olution ▁3 ▁( un it ▁rate ): ▁The ▁area ▁covered ▁under ▁the ▁small ▁sw ing ▁set ▁is ▁9 x ▁12 =10 8 ▁square ▁feet .
▁The ▁number ▁of ▁pages ▁she ▁has ▁finished ▁is ▁x ▁+ ▁90, ▁which ▁is ▁90 .00 00 00 00 00 000 3 ▁+ ▁90 ▁= ▁180 .00 00 00 00 00 000 6 ▁pages .
▁There ▁are ▁many ; ▁see , ▁for ▁example
▁2. ▁Div iding ▁the ▁b ars ▁equally ▁between ▁Thomas ▁and ▁his ▁8 ▁friends ▁(9 ▁people ▁in ▁total ), ▁but ▁Peter ▁and ▁Al ice ▁get ▁3 ▁extra ▁b ars ▁each :
▁6. ▁Paul ▁and ▁Pol ly ▁take ▁b ars ▁for ▁their ▁club s :
▁Therefore , ▁the ▁maximum ▁marks ▁for ▁P ap er ▁A , ▁P ap er ▁B , ▁and ▁P ap er ▁C ▁are ▁16 0, ▁15 0, ▁and ▁200 , ▁respectively .
▁Work ▁done ▁in ▁14 ▁days ▁= ▁4 20 ▁work ▁units
▁If ▁K ir an ▁can ▁fin ish ▁the ▁remaining ▁2 /3 ▁of ▁the ▁work ▁in ▁12 ▁days , ▁we ▁can ▁calculate ▁the ▁rate ▁at ▁which ▁he ▁works ▁per ▁day ▁for ▁the ▁remaining ▁work .
▁60 ▁gra p es ▁+ ▁30 ▁stra w ber ries ▁+ ▁80 ▁or anges ▁= ▁17 0 ▁f ruits ▁in ▁total .
▁Next , ▁let ' s ▁calculate ▁the ▁total ▁st or age ▁space ▁used ▁by ▁the ▁video ▁f iles :
▁a ▁rectangle ▁that ▁is ▁not ▁a ▁square
▁If ▁we ▁take ▁the ▁height ▁of ▁all ▁the ▁boxes ▁small ▁enough , ▁we ▁get ▁a ▁very ▁good ▁approxim ation ▁of ▁the ▁volume ▁of ▁the ▁figure ▁above .
▁In ▁Figure ▁19 ▁have ▁drawn ▁the ▁points ▁and ▁represented ▁the ▁line ▁jo ining ▁them ▁by ▁d ot ted ▁lines .
▁Therefore , ▁there ▁are ▁24 ▁smaller ▁cub es ▁that ▁are ▁pain ted ▁on ▁only ▁two ▁faces .
▁Since ▁each ▁man ▁contrib utes ▁1 ▁work ▁unit ▁per ▁day , ▁we ▁need ▁30 ▁men ▁to ▁continue ▁working ▁for ▁the ▁remaining ▁26 ▁days ▁to ▁complete ▁the ▁work .
▁Now ▁let ' s ▁look ▁at ▁two ▁line ▁se gments .
▁A ▁plane ▁that ▁is ▁per pend ic ular ▁to ▁a ▁face , ▁but ▁not ▁par alle l ▁to ▁any ▁face ▁will ▁cut ▁the ▁cub e ▁in ▁a ▁rectangle ▁that ▁is ▁not ▁a ▁square .
▁Yet , ▁they ▁these ▁are ▁different ▁types ▁of ▁measure ments , ▁and ▁they ▁are ▁expressed ▁using ▁different ▁units , ▁and ▁are ▁calculated ▁in ▁two ▁different ▁ways .
▁The ▁store ▁used ▁9 ▁shel ves ▁to ▁put ▁the ▁remaining ▁col oring ▁books ▁with ▁6 ▁on ▁each ▁she lf .
▁If ▁she ▁spent ▁4 ▁minutes ▁on ▁each ▁piece ▁of ▁furn iture , ▁then ▁the ▁total ▁time ▁spent ▁is ▁10 ▁pieces ▁* ▁4 ▁minutes / p ie ce ▁= ▁40 ▁minutes .
▁Another ▁key ▁re vel ation ▁is ▁that ▁rect angular ▁pr ism s ▁with ▁different ▁d imens ions ▁can ▁have ▁the ▁same ▁volume .
▁In ▁this ▁case , ▁a ▁in ▁the ▁typ ical ▁form ▁a 2 ▁+ ▁2 ab ▁+ ▁b 2 ▁rep res ents ▁5 ▁tens , ▁that ▁is , ▁50 , ▁and ▁b ▁rep res ents ▁7.
▁The ▁total ▁length ▁of ▁the ▁train s ▁is ▁350 ▁m ▁+ ▁450 ▁m ▁+ ▁250 ▁m ▁= ▁10 50 ▁m .
▁She ▁bought ▁5 ▁fewer ▁blue ▁but t ons ▁than ▁the ▁green ▁but t ons , ▁so ▁she ▁bought ▁90 ▁- ▁5 ▁= ▁85 ▁blue ▁but t ons .
▁- ▁Am oura ▁was ▁55 ▁minutes ▁late .
▁Therefore , ▁the ▁number ▁of ▁people ▁who ▁play ▁k ab add i ▁only ▁is :
▁To ▁find ▁the ▁percent ▁of ▁the ▁population ▁that ▁are ▁employ ed ▁mal es , ▁we ▁divide ▁the ▁number ▁of ▁employ ed ▁mal es ▁by ▁the ▁total ▁population ▁and ▁then ▁multiply ▁by ▁100 :
▁Now , ▁if ▁we ▁use ▁all ▁three ▁bu cket s ▁together ▁in ▁each ▁turn , ▁the ▁total ▁capacity ▁filled ▁in ▁one ▁turn ▁would ▁be ▁the ▁sum ▁of ▁the ▁cap ac ities ▁of ▁bu cket s ▁P , ▁Q , ▁and ▁R :
▁Do ▁the ▁same ▁for ▁triangle ▁A ' B ' C ' ".
▁( a ) ▁lands ▁on ▁the ▁d as hed ▁rectangle ▁in ▁figure
▁Total ▁blue ▁and ▁red ▁ball oons ▁= ▁Total ▁blue ▁ball oons ▁+ ▁Total ▁red ▁ball oons
▁If ▁the ▁delivery ▁dri ver ▁has ▁to ▁drop ▁off ▁9 ▁boxes ▁at ▁each ▁stop ▁and ▁he ▁has ▁27 ▁boxes ▁in ▁total , ▁we ▁can ▁calculate ▁the ▁number ▁of ▁st ops ▁he ▁has ▁to ▁make ▁by ▁div iding ▁the ▁total ▁number ▁of ▁boxes ▁by ▁the ▁number ▁of ▁boxes ▁dropped ▁off ▁at ▁each ▁stop .
▁If ▁the ▁faces ▁are ▁all ▁equ ilater al ▁triangles , ▁it ▁s ▁a ▁regular ▁t et ra hed ron .
▁Since ▁x ▁must ▁be ▁an ▁integer , ▁the ▁largest ▁integer ▁less ▁than ▁or ▁equal ▁to ▁3 .0 3 ▁is ▁3.
▁Total ▁yellow ▁ball oons ▁= ▁S andy ' s ▁yellow ▁+ ▁Ch ris ' s ▁yellow
▁This ▁is ▁st rict ly ▁true ▁for ▁rect angles : ▁perimeter ▁is ▁the ▁sum ▁of ▁all ▁of ▁the ▁sides , ▁and ▁area ▁is ▁found ▁by ▁multiply ing ▁the ▁two ▁side ▁lengths .
▁Am it ▁worked ▁for ▁3 ▁days , ▁so ▁the ▁amount ▁of ▁work ▁he ▁completed ▁is :
▁One ▁example ▁is ▁to ▁cut ▁through ▁the ▁top ▁face ▁at ▁a ▁corn er ▁and ▁a ▁mid point ▁of ▁a ▁non - ad j ac ent ▁side , ▁and ▁cut ▁to ▁a ▁different ▁corn er ▁and ▁mid point ▁in ▁the ▁bott om ▁face .
▁5. ▁If ▁we ▁f old ▁a ▁piece ▁of ▁paper ▁once , ▁we ▁get ▁2 ▁smaller ▁pieces .
▁This ▁experiment ▁is ▁conv inc ing ▁that ▁the ▁statement ▁is ▁true , ▁but ▁does ▁not ▁tell ▁us ▁why ▁it ▁is ▁true .
▁- ▁B ars ▁taken ▁by ▁Paul ' s ▁club ▁= ▁9
▁The ▁perimeter ▁of ▁any ▁poly gon al ▁region ▁is ▁the ▁sum ▁of ▁the ▁lengths ▁of ▁its ▁sides .
▁If ▁the ▁first ▁term ▁and ▁common ▁ratio ▁are ▁given , ▁or ▁if ▁any ▁two ▁terms ▁are ▁given , ▁we ▁can ▁find ▁the ▁series .
▁So , ▁the ▁selling ▁price ▁for ▁A ▁( which ▁is ▁the ▁cost ▁price ▁for ▁B ) ▁= ▁Cost ▁price ▁for ▁A ▁+ ▁Prof it ▁for ▁A
▁She ▁gives ▁each ▁friend ▁1/ 6 ▁of ▁her ▁or anges , ▁so ▁each ▁friend ▁gets :
▁We ▁subtract ▁27 ▁from ▁4 2, ▁and ▁bring ▁down ▁the ▁next ▁group , ▁87 5.
▁- ▁For ▁n = 2, ▁the ▁fraction ▁is ▁14 / 16 .
▁Now ▁measure ▁the ▁horizontal ▁and ▁vertical ▁line ▁se gments ▁( the ▁d as hed ▁line ▁se gments ▁in ▁the ▁figure ), ▁to ▁confirm ▁the ▁Py th ag ore an ▁the ore m .
▁Any ▁two ▁points ▁or ▁angles ▁are ▁similar ; ▁because ▁they ▁are ▁con g ru ent .
▁All en ' s ▁pages ▁+ ▁Charles ' s ▁pages ▁+ ▁Le x ' s ▁pages ▁+ ▁She ila ' s ▁pages
▁- ▁Employ ees ▁with ▁over time : ▁4 ▁employees ▁x ▁3 ▁minutes ▁each ▁= ▁12 ▁minutes
▁What ▁is ▁the ▁area ▁of ▁the ▁material ▁necessary ▁to ▁cover ▁the ▁surface ▁of ▁the ▁box .
▁Let ' s ▁calculate ▁the ▁total ▁percentage ▁of ▁his ▁salary ▁that ▁he ▁spe nds ▁on ▁various ▁expenses :
▁But ▁since ▁we ▁can ' t ▁have ▁half ▁a ▁bar , ▁let ' s ▁adjust ▁it ▁to ▁37 ▁b ars ▁each ▁and ▁1 ▁bar ▁will ▁be ▁left ▁un acc ount ed ▁for .
▁These ▁pre con ceived ▁conditions ▁when ▁accur ately ▁form ulated ▁may ▁be ▁called ▁ax i oms ▁of ▁quant ity .
▁50% ▁of ▁44 ▁c ows ▁= ▁0 .50 ▁* ▁44 ▁= ▁22 ▁female ▁c ows
▁This ▁is ▁because ▁rigid ▁mot ions ▁and ▁dil ations ▁do ▁not ▁change ▁the ▁measure ▁of ▁angles .
▁The ▁last ▁two ▁figures ▁of ▁the ▁root ▁are ▁found ▁by ▁division .
▁As ▁the ▁angle ▁at ▁C ▁grow s ▁from ▁very ▁t iny ▁to ▁very ▁near ▁a ▁straight ▁angle , ▁the ▁length ▁of ▁its ▁opp osing ▁side ▁st ead ily ▁increases .
▁In ▁turn , ▁the ▁student ▁should ▁be ▁better ▁able ▁to ▁explain ▁the ▁reason ing ▁behind ▁the ▁Py th ag ore an ▁the ore m , ▁than ▁to ▁provide ▁it ▁in ▁a ▁form ▁that ▁exhib its ▁form ▁over ▁gr asp .
▁If ▁it ▁takes ▁60 ▁turn s ▁for ▁bu cket ▁P ▁to ▁fill ▁the ▁dr um , ▁then ▁the ▁total ▁capacity ▁of ▁the ▁dr um ▁is ▁60 ▁times ▁the ▁capacity ▁of ▁bu cket ▁P , ▁which ▁is ▁60 ▁* ▁3 Q ▁= ▁180 Q .
▁To ▁find ▁the ▁fraction ▁of ▁the ▁book ▁she ▁has ▁finished , ▁we ▁divide ▁the ▁number ▁of ▁pages ▁she ▁has ▁finished ▁by ▁the ▁total ▁number ▁of ▁pages ▁in ▁the ▁book :
▁We ▁are ▁looking ▁for ▁non - ne g ative ▁integer ▁solutions ▁to ▁this ▁equation .
▁An ▁is os ce les ▁right ▁triangle ▁has ▁a ▁hypot en use ▁of ▁length ▁100 ▁cm .
▁Total ▁number ▁of ▁selected ▁film s ▁= ▁Number ▁of ▁selected ▁black - and - wh ite ▁film s ▁+ ▁Number ▁of ▁selected ▁col or ▁film s
▁Total ▁Ben z ene ▁( C 6 H 6) ▁required ▁= ▁3 ▁m oles ▁( for ▁Ch lor ob en z ene )
▁Therefore , ▁the ▁gain ▁percent ▁on ▁the ▁sale ▁of ▁the ▁cycle ▁is ▁60 %.
▁The ▁pe ak ▁of ▁the ▁roof ▁is ▁3 ' ▁above ▁the ▁ce iling ▁of ▁the ▁s hed .
▁Also ▁the ▁tr ig on omet r ical ▁form ul æ ▁quot ed ▁above ▁are ▁other ▁ex amples ▁of ▁the ▁same ▁fact .
▁3. ▁If ▁the ▁py ram id ▁is ▁cut ▁with ▁a ▁plane ▁per pend ic ular ▁to ▁the ▁base ▁and ▁par alle l ▁to ▁one ▁of ▁the ▁ed ges ▁of ▁the ▁base , ▁but ▁not ▁through ▁the ▁top ▁ver te x , ▁the ▁inter section ▁of ▁the ▁py ram id ▁and ▁the ▁plane ▁is ▁an ▁is os ce les ▁tra pe z oid al ▁cross - section .
▁This ▁doesn ' t ▁seem ▁right ▁because ▁if ▁b ' s ▁work ▁rate ▁was ▁0 , ▁b ▁wouldn ' t ▁be ▁contrib uting ▁to ▁the ▁work ▁at ▁all , ▁which ▁cont r adic ts ▁the ▁initial ▁statement ▁that ▁a ▁and ▁b ▁can ▁fin ish ▁the ▁work ▁together ▁in ▁16 ▁days .
▁Total ▁dem er its ▁so ▁far ▁= ▁44 ▁dem er its
▁Again , ▁without ▁knowing ▁the ▁common ▁ratio , ▁we ▁can ' t ▁determine ▁the ▁exact ▁numbers .
▁The ▁answer ▁is ▁" y es ," ▁and ▁there ▁are ▁many ▁ways ▁to ▁show ▁this .
▁To ▁calculate ▁the ▁amount ▁received ▁by ▁Sam ▁at ▁the ▁end ▁of ▁the ▁year ▁with ▁half - year ly ▁comp ound ing , ▁we ▁need ▁to ▁adjust ▁the ▁interest ▁rate ▁and ▁the ▁number ▁of ▁comp ound ing ▁periods .
▁This ▁gives ▁strong ▁evidence ▁that , ▁among ▁all ▁rect angles ▁of ▁given ▁perimeter , ▁the ▁square ▁has ▁the ▁largest ▁area .
▁Let ' s ▁use ▁the ▁work ▁formula :
▁We ▁can ▁count ▁the ▁factors ▁of ▁3 ▁in ▁each ▁of ▁these ▁numbers :
▁If the ▁pr ism ▁is ▁not ▁a ▁right ▁pr ism , ▁as ▁in ▁Figure ▁28, ▁can ▁we ▁still ▁find ▁its ▁volume ?
▁Therefore , ▁the ▁cash ▁disc ount ▁percentage ▁offered ▁by ▁the ▁deal er ▁is ▁60 %.
▁Now , ▁let ' s ▁denote ▁the ▁transportation ▁charges ▁as ▁" T ".
▁Figure ▁15 : of ▁these ▁triangles , ▁one ▁of ▁which ▁is ▁the ▁image ▁of ▁ABC .
▁Work ▁= ▁Number ▁of ▁workers ▁× ▁Number ▁of ▁days
▁The ▁radi us ▁of ▁the ▁se m ic irc le ▁at ▁the ▁ends ▁formed ▁by ▁the ▁inside ▁of ▁the ▁the ▁second ▁l ane ▁is ▁1.2 ▁m ▁longer ▁than ▁the ▁inside ▁length ▁of ▁the ▁first ▁l ane , ▁so ▁is ▁31 .8 6 +1 .2 ▁= ▁33 .06 ▁m .
▁1. ▁The ▁total ▁cost ▁including ▁tax ▁is ▁$ 82 7.
▁To ▁complete ▁our ▁list ▁of ▁fundamental ▁figures , ▁we ▁include ▁the ▁circle , ▁discussed ▁in ▁Chapter ▁5.
▁So , ▁the ▁committee ▁select s ▁y /5 ▁black - and - wh ite ▁film s .
▁The ▁f ence ▁can ▁hold ▁a ▁maximum ▁of ▁20 ▁l bs .
▁For ▁example , ▁as ▁we ▁saw ▁above , ▁a ▁cub e ▁can ▁be ▁view ed ▁as ▁a ▁st ack ▁of ▁squ ares , ▁all ▁of ▁which ▁have ▁the ▁same ▁side ▁lengths .
▁So , ▁the ▁total ▁st or age ▁space ▁used ▁on ▁Am y ' s ▁fl ash ▁drive ▁after ▁down lo ading ▁the ▁picture ▁f iles ▁is ▁11 16 ▁M B .
▁To ▁solve ▁this ▁problem , ▁we ▁need ▁to ▁consider ▁the ▁relative ▁speed s ▁of ▁the ▁train s ▁as ▁they ▁approach ▁each ▁other ▁and ▁the ▁total ▁distance ▁they ▁need ▁to ▁cover ▁to ▁be ▁clear ▁of ▁each ▁other .
▁This ▁is ▁not ▁the ▁usual ▁section ▁of ▁a ▁b age l , ▁which ▁will ▁be ▁along ▁the ▁plane ▁of ▁its ▁major ▁diam eter .
▁- ▁Water : ▁5 ▁kg ▁( rem aining ▁after ▁ev ap or ation ) ▁+ ▁1.4 ▁kg ▁( add ed ) ▁= ▁6 .4 ▁kg
▁Let ' s ▁first ▁find ▁out ▁how ▁many ▁black - and - wh ite ▁film s ▁are ▁selected ▁by ▁the ▁committee .
▁Let ' s ▁first ▁find ▁out ▁how ▁many ▁pages ▁P ra k ash ▁can ▁copy ▁in ▁1 ▁hour .
▁This ▁allow s ▁the ▁cut ▁to ▁form ▁no ▁90 deg ▁in ▁angles .
▁Now , ▁if ▁the ▁long ▁side ▁of ▁the ▁rectangle ▁is ▁15 ▁units , ▁the ▁sh or ter ▁side ▁has ▁length ▁10 ▁units , ▁and ▁the ▁area ▁is ▁150 ▁square ▁units .
▁2. ▁[ 2 ▁/ ▁3 ] ▁is ▁the ▁greatest ▁integer ▁less ▁than ▁or ▁equal ▁to ▁2 / 3, ▁which ▁is ▁0 ▁because ▁2 /3 ▁is ▁less ▁than ▁1.
▁Total ▁st or age ▁space ▁used ▁= ▁20 ▁M B ▁+ ▁10 50 ▁M B ▁+ ▁46 ▁M B
▁So , ▁the ▁area ▁of ▁the ▁entire ▁figure ▁is ▁4 h ^{2} + h ^{2} =5 h ^{2} , ▁and ▁we ▁are ▁given ▁that ▁that ▁is ▁12 50 ▁s q .
▁We ▁can ▁calculate ▁the ▁percentage ▁of ▁eth an ol ▁as ▁follows :
▁Note ▁that ▁sections ▁of ▁the ▁con e ▁by ▁a ▁plane ▁par alle l ▁to ▁the ▁base ▁produ ces ▁a ▁figure ▁that ▁is ▁a ▁scaled ▁version ▁of ▁the ▁base , ▁with ▁the ▁scale ▁factor ▁reducing ▁as ▁we ▁move ▁toward ▁the ▁a pe x .
▁- ▁Per cent ▁of ▁employ ed ▁mal es ▁= ▁(2 1 ▁/ ▁100 ) ▁* ▁100
▁Consider ▁the ▁qu adr ilater al ▁in ▁Figure ▁5.
▁Therefore , ▁S her ry ' s ▁rec ipe ▁makes ▁enough ▁bat ter ▁for ▁3 ▁lo aves .
▁Equ ation ▁2 ▁( P r ice ▁difference ▁between ▁A ▁and ▁B ):
▁Then ▁the ▁actual ▁distance ▁in ▁km ▁is
▁The ▁P ent ag on , ▁the ▁head qu ar ters ▁of ▁the ▁U . S . ▁Department ▁of ▁Defense , ▁is ▁a ▁regular ▁five - s ided ▁figure ▁with ▁a ▁total ▁of ▁6 .5 ▁million ▁square ▁feet ▁of ▁floor ▁space ▁on ▁seven ▁levels , ▁two ▁of ▁which ▁are ▁under ground .
▁In ▁order ▁to ▁see ▁why ▁this ▁is ▁true , ▁we ▁show ▁how ▁to ▁draw ▁all ▁triangles ▁with ▁two ▁side ▁lengths ▁a ▁and ▁b .
▁Car rie ▁brought ▁5 ▁p ounds ▁less ▁than ▁the ▁amount ▁of ▁flour ▁We nd i ▁brought , ▁so :
▁Se lling ▁price ▁= ▁150 % ▁of ▁( T otal ▁cost ▁including ▁transportation )
▁Also , ▁since ▁2 ab ▁+ ▁b 2 ▁= ▁(2 a ▁+ ▁b ) b , ▁the ▁divis or ▁is ▁completed ▁by ▁adding ▁to ▁the ▁trial - d iv is or ▁the ▁new ▁term ▁of ▁the ▁root .
▁In ▁other ▁words , ▁the ▁square ▁root ▁of ▁a ▁number ▁expressed ▁by ▁one ▁or ▁two ▁figures ▁is ▁a ▁number ▁of ▁one ▁figure , ▁of ▁a ▁number ▁expressed ▁by ▁three ▁or ▁four ▁figures ▁is ▁a ▁number ▁of ▁two ▁figures , ▁and ▁so ▁on .
▁St art ▁a ▁plane ▁in ▁three - d imens ions ▁and ▁a ▁point ▁A ▁not ▁on ▁the ▁plane , ▁and ▁a ▁poly gon al ▁figure ▁F ▁on ▁the ▁plane .
▁Now , ▁let ' s ▁add ▁the ▁minutes ▁Ind ira ▁played ▁cricket ▁to ▁the ▁total ▁minutes ▁Se an ▁played ▁to ▁find ▁out ▁how ▁many ▁minutes ▁they ▁played ▁together .
▁The ▁new ▁ratio ▁of ▁board ers ▁to ▁day ▁scholars ▁is ▁1 ▁to ▁2, ▁which ▁we ▁can ▁write ▁as ▁( B ▁+ ▁80 ) / D ▁= ▁1/ 2.
▁Not ice ▁that ▁we ▁do ▁not ▁require ▁that ▁the ▁figures ▁have ▁the ▁same ▁size ▁and ▁shape , ▁only ▁that ▁they ▁have ▁the ▁same ▁area .
▁If ▁Kar le e ▁has ▁3 /5 ▁as ▁many ▁stra w ber ries ▁as ▁gra p es , ▁and ▁she ▁has ▁100 ▁gra p es , ▁then ▁she ▁has :
▁Although ▁this ▁is ▁a ▁" pro of ▁without ▁words ," ▁here ▁is ▁a ▁des cript ion ▁of ▁how ▁it ▁goes : ▁For ▁the ▁original ▁square ▁of ▁side ▁length ▁a + b ▁is ▁now ▁sub d iv ided ▁in ▁a ▁different ▁conf ig uration : ▁the ▁bott om ▁left ▁corn er ▁is ▁filled ▁with ▁a ▁square ▁of ▁side ▁length ▁b , ▁and ▁the ▁u pper ▁right ▁corn er , ▁by ▁a ▁square ▁of ▁side ▁length ▁a .
▁Although ▁prim itive , ▁this ▁method ▁is ▁important , ▁because ▁it ▁rel ies ▁directly ▁on ▁the ▁defin ition ▁of ▁volume ▁and ▁therefore ▁emphas izes ▁the ▁meaning ▁of ▁volume .
▁For ▁the ▁inequ ality ▁to ▁hold , ▁3 c ▁must ▁be ▁greater ▁than ▁x .
▁Now , ▁we ▁want ▁the ▁remaining ▁work ▁to ▁be ▁completed ▁in ▁the ▁remaining ▁26 ▁days ▁( since ▁14 ▁days ▁have ▁already ▁passed ▁and ▁the ▁total ▁time ▁is ▁40 ▁days ).
▁In ▁total , ▁the ▁tail or ▁purchased ▁90 ▁green ▁but t ons ▁+ ▁100 ▁yellow ▁but t ons ▁+ ▁85 ▁blue ▁but t ons ▁= ▁2 75 ▁but t ons .
▁So ▁there ▁are ▁11 ▁fractions ▁with ▁a ▁two - d ig it ▁denominator ▁that ▁have ▁the ▁same ▁magn itude ▁as ▁7 / 8.
▁Given ▁the ▁points ▁A ▁and ▁B , ▁draw ▁the ▁rect angular ▁pr ism ▁with ▁sides ▁par alle l ▁to ▁the ▁coordinate ▁plan es ▁that ▁has ▁A ▁and ▁B ▁as ▁diam et r ically ▁opposite ▁vert ices ▁( re fer ▁to ▁Figure ▁16 ).
▁So , ▁the ▁value ▁of ▁d ^2 ▁+ ▁5 d ▁when ▁d ▁= ▁11 ▁is ▁17 6.
▁Therefore , ▁Jack ' s ▁basket ▁can ▁currently ▁hold ▁4 ▁more ▁apples .
▁Now , ▁let ' s ▁see ▁how ▁many ▁te es ▁Bill ▁can ▁get ▁from ▁the ▁gener ic ▁g ol f ▁te es ▁that ▁are ▁pack aged ▁by ▁the ▁doz en .
▁Since ▁the ▁dep th ▁of ▁sand ▁is ▁the ▁same , ▁the ▁amount ▁of ▁sand ▁needed ▁for ▁the ▁large ▁sw ing ▁set ▁is ▁2 .25 ▁times ▁as ▁much ▁as ▁is ▁needed ▁for ▁the ▁small ▁sw ing ▁set , ▁and ▁they ▁will ▁need ▁2 .25 ▁times ▁as ▁many ▁bags .
▁2. ▁Am oura ▁arrived ▁30 ▁minutes ▁later ▁than ▁Paul .
▁First ▁of ▁all , ▁we ▁know ▁the ▁coordin ates ▁for ▁R : ▁( 7, 2).
▁Since ▁each ▁lo af ▁contains ▁12 ▁slic es , ▁we ▁need ▁to ▁determine ▁how ▁many ▁lo aves ▁will ▁be ▁needed ▁to ▁provide ▁15 6 ▁slic es .
▁The ▁team ▁needs ▁to ▁score ▁at ▁a ▁run ▁rate ▁of ▁5 .75 ▁runs ▁per ▁over ▁to ▁reach ▁the ▁target .
▁Total ▁b ars ▁for ▁the ▁other ▁7 ▁people ▁= ▁33 ▁b ars ▁each
▁for ▁a ▁right ▁triangle ▁whose ▁leg ▁lengths ▁are ▁a ▁and ▁b ▁and ▁whose ▁hypot en use ▁is ▁of ▁length ▁c .
▁So , ▁the ▁amount ▁in ▁4 ▁years ▁is ▁Rs . ▁11 20 .
▁Jacob ▁is ▁currently ▁4 ▁years ▁old .
▁A ▁way ▁to ▁find ▁the ▁volume ▁of ▁a ▁sol id ▁shape ▁is ▁to ▁understand ▁how ▁it ▁can ▁be ▁developed ▁out ▁of ▁two ▁d imens ional ▁figures .
▁20 ▁acres ▁* ▁400 ▁gallons / ac re ▁= ▁8 000 ▁gallons
▁The ▁error , ▁12 20, ▁is ▁well ▁within ▁one ▁percent ▁of ▁c ^{2} , ▁so ▁this ▁angle ▁can ▁be ▁taken ▁to ▁be ▁a ▁right ▁angle .
▁We ▁know ▁S ier ra ▁ate ▁12 ▁and ▁Bill ▁ate ▁6, ▁so ▁together ▁they ▁ate ▁12 ▁+ ▁6 ▁= ▁18 ▁p ies .
▁4 ▁days ▁* ▁10 ▁notes / day ▁= ▁40 ▁notes
▁What ▁are ▁the ▁possible ▁surface ▁areas ▁of ▁these ▁pack ages ?
▁There ▁is ▁also ▁a ▁reflection ▁of ▁P ▁to ▁Q : ▁f old ▁the ▁paper ▁in ▁such ▁a ▁way ▁that ▁P ▁lands ▁on ▁top ▁of ▁Q .
▁For ▁a ▁circle , ▁again ▁the ▁answer ▁is ▁" y es " : ▁if ▁you ▁know ▁the ▁perimeter ▁( i . e ., ▁the ▁circum ference ) ▁you ▁can ▁find ▁the ▁radi us ▁and ▁hence ▁the ▁area .
▁The ▁horizontal ▁line ▁is ▁30 ▁mm , ▁and ▁the ▁vertical ▁line ▁is ▁27 ▁mm .
▁So , ▁the ▁total ▁cost ▁before ▁transportation ▁charges ▁is :
▁If ▁the ▁least ▁number ▁to ▁be ▁added ▁to ▁10 52 ▁to ▁make ▁it ▁completely ▁divis ible ▁by ▁a ▁certain ▁number ▁is ▁6, ▁then ▁the ▁sum ▁that ▁is ▁divis ible ▁by ▁the ▁divis or ▁is ▁10 52 ▁+ ▁6 ▁= ▁10 5 8.
▁Let ▁the ▁lengths ▁of ▁the ▁par alle l ▁sides ▁be ▁b 1 ▁and ▁b 2, ▁and ▁the ▁distance ▁between ▁them ▁h .
▁Since ▁a ▁cub e ▁has ▁6 ▁faces , ▁a ▁plane ▁section ▁can ▁have ▁at ▁most ▁6 ▁sides ; ▁so ▁this ▁ans wers ▁part ▁h ▁in ▁the ▁negative : ▁no ▁o ct ag ons ▁( and ▁in ▁fact , ▁no ▁seven - s ided ▁poly g ons ▁either ).
▁Even ▁today , ▁we ▁have ▁no ▁satisf act ory ▁way ▁to ▁visual ize ▁putting ▁three ▁circular ▁con es ▁of ▁the ▁same ▁size ▁and ▁shape ▁into ▁the ▁circum sc rib ed ▁cyl inder .
▁The ▁name ▁poly hed ron ▁comes ▁from ▁the ▁Greek ; ▁poly ▁meaning ▁" m any " ▁and ▁he d ron ▁meaning ▁" f ace ," ▁so ▁poly hed ron ▁liter ally ▁means ▁" m any ▁faces ."
▁Since ▁a ▁is ▁3 ▁tens ▁or ▁30, ▁3 a 2 ▁= ▁3 ▁× ▁30 2 ▁, ▁or ▁27 00.
▁The ▁volume ▁of ▁a ▁pr ism ▁is ▁the ▁product ▁of ▁the ▁height ▁by ▁the ▁area ▁of ▁the ▁base .
▁1. ▁The ▁number ▁is ▁dou bled : ▁2 x
▁To ▁calculate ▁the ▁sa v ings ▁per ▁b age l ▁when ▁buy ing ▁a ▁doz en ▁at ▁a ▁time , ▁we ▁first ▁need ▁to ▁determine ▁the ▁cost ▁per ▁b age l ▁when ▁buy ing ▁them ▁ind iv id ually ▁and ▁then ▁when ▁buy ing ▁a ▁doz en .
▁Then ▁the ▁equivalent ▁fraction ▁would ▁be ▁(7 n ) / ( 8 n ).
▁1. ▁We ▁know ▁that ▁one ▁not ep ad ▁last s ▁for ▁4 ▁days ▁with ▁10 ▁notes ▁written ▁per ▁day .
▁What ▁is ▁the ▁length ▁of ▁the ▁long est ▁line ▁segment ▁in ▁the ▁unit ▁cub e ?
▁Now ▁draw ▁a ▁circle ▁with ▁center ▁at ▁C ▁and ▁radi us ▁of ▁length ▁b ▁as ▁in ▁Figure ▁16 .
▁If ▁we ▁cover ▁B ▁with ▁a ▁gr id , ▁we ▁can ▁draw ▁out ▁that ▁gr id ▁along ▁the ▁par alle l ▁lines , ▁getting ▁a ▁dec om p osition ▁of ▁that ▁region ▁into ▁rect angular ▁pr ism s , ▁the ▁base ▁of ▁each ▁is ▁a ▁rectangle ▁in ▁the ▁plan ar ▁gr id .
▁Now ▁we ▁have ▁two ▁expressions ▁for ▁SP , ▁one ▁in ▁terms ▁of ▁C P ▁and ▁one ▁in ▁terms ▁of ▁MP .
▁So , ▁Bruce ▁purchased ▁9 ▁kg ▁of ▁mango es .
▁This ▁means ▁that ▁the ▁interest ▁earned ▁in ▁5 ▁years ▁is ▁Rs . ▁1200 ▁- ▁P . ▁Since ▁we ▁are ▁given ▁that ▁the ▁principal ▁sum ▁P ▁is ▁Rs . ▁800 , ▁we ▁can ▁write :
▁Work ▁done ▁in ▁14 ▁days ▁= ▁Number ▁of ▁men ▁× ▁Number ▁of ▁days
▁Similarly , ▁since ▁the ▁trad er ▁l oses ▁13% ▁on ▁the ▁second ▁car , ▁we ▁can ▁write :
▁The ▁statement ▁" 3 ^ k ▁- ▁k ^3 ▁= ▁1 " ▁is ▁a ▁condition ▁that ▁must ▁be ▁satisfied ▁for ▁some ▁non - ne g ative ▁integer ▁k .
▁We ▁shall ▁be ▁considering ▁con es ▁with ▁the ▁property ▁that ▁the ▁line ▁from ▁the ▁a pe x ▁to ▁the ▁center ▁of ▁the ▁base ▁is ▁per pend ic ular ▁to ▁the ▁base ▁( te chn ically , ▁these ▁are ▁called ▁right ▁con es ).
▁Thus ▁lengths ▁are ▁quant ities ▁of ▁the ▁same ▁kind , ▁so ▁are ▁areas , ▁and ▁so ▁are ▁vol um es .
▁Since ▁we ▁just ▁want ▁an ▁approxim ate ▁answer , ▁we ▁look ▁for ▁the ▁integer ▁whose ▁square ▁is ▁close ▁to ▁26 0 : ▁that ▁would ▁be ▁16 ▁(16 ^{2} = 25 6 ).
▁Give ▁an ▁approxim ate ▁value ▁for ▁the ▁length ▁of ▁its ▁hypot en use .
▁If ▁K im ▁now ▁has ▁6 ▁hours ▁of ▁classes ▁per ▁day ▁after ▁dro pping ▁1 ▁class , ▁and ▁each ▁class ▁last s ▁2 ▁hours , ▁then ▁she ▁currently ▁has ▁6 ▁hours ▁/ ▁2 ▁hours ▁per ▁class ▁= ▁3 ▁classes .
▁The ▁coordinate ▁rule ▁for ▁the ▁rot ation ▁by ▁90 degrees ▁is ▁( x , y )- > ( y , x ).
▁We ▁are ▁looking ▁for ▁two - d ig it ▁denomin ators , ▁so ▁8 n ▁must ▁be ▁a ▁two - d ig it ▁number .
▁The ▁most ▁basic ▁way ▁to ▁determine ▁the ▁volume ▁of ▁a ▁sol id ▁shape ▁is ▁to ▁make ▁the ▁shape ▁out ▁of ▁cub es ▁( f ill ing ▁the ▁inside ▁completely ) ▁and ▁to ▁count ▁how ▁many ▁cub es ▁it ▁took .
▁Pl ane ▁sections ▁provide ▁two - d imens ional ▁information ▁about ▁the ▁inside ▁of ▁a ▁shape .
▁To ▁determine ▁the ▁total ▁three - point ▁area ▁for ▁Te am ▁A , ▁we ▁subtract ▁the ▁total ▁area ▁of ▁the ▁se m ic irc le ▁from ▁the ▁total ▁area ▁of ▁the ▁rectangle :
▁We ▁are ▁given ▁that ▁for ▁each ▁Rs . ▁a ▁has , ▁b ▁has ▁65 ▁pa isa .
▁First , ▁draw ▁a ▁triangle ▁with ▁a ▁horizontal ▁base ▁( the ▁triangle ▁with ▁sol id ▁sides ▁in ▁Figure ▁3 ).
▁Total ▁volume ▁= ▁Or ig inal ▁gas oh ol ▁volume ▁+ ▁Add ed ▁eth an ol ▁volume
▁Finally , ▁let ' s ▁add ▁up ▁all ▁the ▁f ruits ▁Kar le e ▁is ▁left ▁with :
▁- ▁From ▁3, ▁we ▁get ▁1 ▁factor ▁of ▁3.
▁7 80 ▁work ▁units ▁= ▁M ▁men ▁× ▁26 ▁days
▁Since ▁they ▁select ▁y / x % ▁of ▁the ▁black - and - wh ite ▁film s , ▁we ▁can ▁write ▁this ▁as :
▁H alf way ▁between ▁them ▁is ▁130 ▁/ ▁2 ▁= ▁65 ▁minutes ▁after ▁Paul ' s ▁arrival .
▁A ▁circle ▁graph , ▁also ▁known ▁as ▁a ▁pie ▁ch art , ▁rep res ents ▁100% ▁of ▁the ▁data ▁by ▁a ▁full ▁circle , ▁which ▁is ▁3 60 ▁degrees .
▁But ▁each ▁angle ▁in ▁this ▁expression ▁is ▁supplementary ▁to ▁the ▁correspond ing ▁angle ▁of ▁the ▁triangle , ▁that ▁is , ▁the ▁sum ▁of ▁the ▁measures ▁of ▁the ▁angle ▁is ▁180 degrees .
▁By ▁transport ing ▁the ▁foot - ru le ▁from ▁place ▁to ▁place ▁we ▁judge ▁of ▁the ▁equ ality ▁of ▁lengths .
▁Since ▁line ▁se gments ▁of ▁the ▁same ▁length ▁are ▁con g ru ent , ▁we ▁can ▁move ▁a ▁on to ▁a ' ▁by ▁a ▁rigid ▁motion .
▁Ch o ose ▁one ▁side ▁of ▁the ▁par alle lo gram , ▁call ▁it ▁the ▁base ▁and ▁its ▁length ▁b , ▁and ▁let ▁h ▁be ▁the ▁distance ▁between ▁the ▁base ▁and ▁its ▁opposite ▁side .
▁If ▁correspond ing ▁sides ▁of ▁triangles ▁ABC ▁and ▁A ' B ' C ' ▁are ▁par alle l , ▁then ▁the ▁triangles ▁are ▁similar .
▁The ▁equ il ib ri um ▁constant ▁K ▁for ▁the ▁first ▁re action ▁cannot ▁be ▁determined ▁without ▁additional ▁information ▁about ▁the ▁concent r ations ▁at ▁equ il ib ri um .
▁Now , ▁let ' s ▁find ▁out ▁how ▁many ▁pages ▁Sub ash ▁and ▁P ra k ash ▁can ▁copy ▁together ▁in ▁1 ▁hour .
▁Since ▁two ▁circ les ▁intersect ▁in ▁at ▁most ▁two ▁points , ▁the ▁circle ▁of ▁radi us ▁c ▁cent ered ▁at ▁B ▁intersect s ▁our ▁circle ▁either ▁in ▁the ▁point ▁A , ▁or ▁the ▁point ▁of ▁reflection ▁of ▁A ' ▁in ▁the ▁line ▁BC .
▁The ▁distance ▁travel ed ▁by ▁each ▁train ▁can ▁be ▁found ▁using ▁the ▁formula :
▁Inc lud ing ▁the ▁apples ▁handed ▁out ▁to ▁students , ▁the ▁ca f eter ia ▁initially ▁had ▁56 ▁apples ▁for ▁p ies ▁+ ▁30 ▁apples ▁for ▁students ▁= ▁86 ▁apples .
▁Let ' s ▁re - e val u ate ▁the ▁given ▁information .
▁So , ▁the ▁half - year ly ▁interest ▁rate ▁is ▁10% ▁/ ▁2 ▁= ▁5 %.
▁Now , ▁as ▁for ▁triangle ▁IV , ▁we ▁can ▁transl ate ▁the ▁right ▁angle ▁of ▁triangle ▁I ▁to ▁that ▁of ▁triangle ▁IV .
▁If ▁the ▁a 2 ▁is ▁sub tra cted ▁from ▁the ▁given ▁expression , ▁the ▁remainder ▁is ▁2 ab ▁+ ▁b 2 ▁.
▁A ▁room ▁is ▁in ▁the ▁shape ▁of ▁a ▁rectangle ▁of ▁wid th ▁12 ▁feet , ▁length ▁20 ▁feet , ▁and ▁height ▁8 ▁feet .
▁She ▁gave ▁her ▁mother ▁65 ▁to w els , ▁so ▁she ▁was ▁left ▁with ▁( G ▁+ ▁4 4) ▁- ▁65 ▁to w els .
▁Since ▁42 8 75 ▁has ▁two ▁groups , ▁the ▁root ▁will ▁have ▁two ▁figures .
▁T im ▁placed ▁14 ▁rulers ▁in ▁the ▁dra wer .
▁Ad am ▁ate ▁3 ▁more ▁p ies ▁than ▁Bill .
▁Therefore , ▁the ▁car ▁t ire ▁will ▁make ▁approximately ▁7 75 ▁revol utions ▁if ▁the ▁car ▁travel s ▁1/ 2 ▁m ile .
▁Now ▁we ▁need ▁to ▁convert ▁the ▁total ▁inches ▁to ▁feet .
▁From ▁the ▁probability ▁of ▁getting ▁at ▁least ▁one ▁contract , ▁we ▁know ▁that :
▁We ▁must ▁be ▁careful ▁to ▁have ▁the ▁last ▁group ▁on ▁the ▁right ▁of ▁the ▁decimal ▁point ▁cont ain ▁three ▁figures , ▁an nex ing ▁c ip hers ▁when ▁necessary .
▁Now ▁we ▁have ▁the ▁height ▁and ▁wid th ▁of ▁the ▁Great ▁Py ram id ▁of ▁G iz a .
▁G rap es ▁given ▁away : ▁20 ▁* ▁2 ▁= ▁40 ▁gra p es .
▁- ▁4 ▁fin ches , ▁so ▁they ▁eat ▁4 ▁* ▁( P / 2) ▁= ▁2 P ▁grams ▁a ▁day ▁together .
▁Now ▁draw ▁the ▁se m ic irc le ▁whose ▁center ▁is ▁C ▁and ▁whose ▁radi us ▁is ▁b ▁( see ▁Figure ▁12 ).
▁Let ' s ▁assume ▁that ▁together , ▁Ar un ▁and ▁T ar un ▁can ▁complete ▁the ▁work ▁in ▁x ▁days .
▁A th le te ▁4 ▁finished ▁25 ▁seconds ▁less ▁than ▁ath le te ▁1, ▁so ▁ath le te ▁4 ▁ran ▁for ▁55 ▁- ▁25 ▁= ▁30 ▁seconds .
▁2. ▁If ▁you ▁comb ine ▁( a ▁fin ite ▁number ▁of ) ▁sh ap es ▁without ▁over l app ing ▁them , ▁then ▁the ▁area ▁of ▁the ▁resulting ▁shape ▁is ▁the ▁sum ▁of ▁the ▁areas ▁of ▁the ▁individual ▁sh ap es .
▁Well , ▁then ▁the ▁translation ▁T ▁takes ▁the ▁angle ▁AVB ▁to ▁the ▁angle ▁A ' V ' B ', ▁so ▁they ▁have ▁the ▁same ▁measure .
▁Since ▁the ▁cub e ▁has ▁12 ▁ed ges ▁in ▁total , ▁we ▁would ▁initially ▁think ▁to ▁multiply ▁4 ▁by ▁12.
▁Now ▁we ▁turn ▁to ▁sh ap es ▁in ▁3 D ▁and ▁how ▁to ▁visual ize ▁them ▁using ▁our ▁knowledge ▁about ▁2 D ▁figures .
▁It ▁will ▁be ▁seen ▁from ▁the ▁groups ▁of ▁figures ▁that ▁the ▁root ▁will ▁have ▁one ▁integral ▁and ▁two ▁decimal ▁places .
▁R ot ate ▁the ▁tra pe z oid ▁around ▁the ▁ver te x ▁V ▁for ▁180 deg , ▁as ▁shown ▁by ▁the ▁red ▁circular ▁ar row .
▁Bill ▁still ▁needs ▁56 ▁te es ▁after ▁purch asing ▁the ▁gener ic ▁te es .
▁So ▁the ▁marked ▁price ▁is ▁fixed ▁at ▁approximately ▁32 .01 % ▁above ▁the ▁cost ▁price .
▁L ast ly , ▁rect angular ▁pr ism s ▁with ▁the ▁same ▁volume ▁can ▁have ▁different ▁surface ▁area .
▁Note : ▁This ▁problem ▁may ▁be ▁beyond ▁the ▁sc ope ▁of ▁8 th ▁grade ▁mathematics , ▁but ▁it ▁still ▁may ▁be ▁worth ▁discuss ing , ▁since ▁it ▁illust rates ▁how ▁the ▁inter action ▁of ▁geomet ry ▁and ▁algebra ▁works ▁to ▁solve ▁a ▁complex ▁struct ural ▁problem .
▁This ▁trial - d iv is or ▁is ▁cont ained ▁5 ▁times ▁in ▁15 8 7 5.
▁4. ▁D il an ▁was ▁10 ▁minutes ▁earlier ▁than ▁In gr id .
▁Luc ille ▁will ▁need ▁to ▁round ▁up ▁to ▁the ▁ne arest ▁whole ▁can , ▁so ▁she ▁will ▁need ▁18 ▁can s ▁of ▁yellow ▁paint .
▁We ▁focus ▁on ▁sections ▁by ▁plan es ▁par alle l ▁to ▁the ▁base ▁of ▁the ▁figure .
▁6. ▁She ▁l oses ▁5 ▁se as he lls , ▁so ▁she ▁now ▁has ▁56 ▁- ▁5 ▁= ▁51 ▁se as he lls .
▁and ▁the ▁area ▁of ▁a ▁side ▁is ▁60 ▁s q .
▁This ▁suggests ▁that , ▁when ▁we ▁are ▁given ▁the ▁face ▁areas , ▁we ▁can ▁solve ▁for ▁the ▁ed ge ▁lengths . This ▁is ▁in ▁fact ▁true : ▁Let ▁a , b , c ▁be ▁the ▁side ▁lengths , ▁and ▁A , B , C ▁the ▁given ▁face ▁areas .
▁Total ▁cost ▁before ▁transportation ▁= ▁Rs ▁16 000
▁This ▁trial ▁divis or ▁is ▁8 6, ▁700 , ▁and ▁if ▁we ▁add ▁3 ab ▁+ ▁b 2 ▁to ▁complete ▁the ▁divis or , ▁when ▁b ▁= ▁1, ▁the ▁complete ▁divis or ▁will ▁be ▁8 6, ▁700 ▁+ ▁5 11 ▁= ▁8 7, ▁21 1, ▁and ▁this ▁is ▁larger ▁than ▁the ▁divide nd ▁8 7, ▁000 .
▁The ▁coordinate ▁rule ▁for ▁the ▁reflection ▁in ▁the ▁line ▁x = y ▁is ▁( x , y )- > ( y , x ).
▁When ▁working ▁together , ▁the ▁rates ▁of ▁the ▁two ▁pr in ters ▁are ▁add itive .
▁So , ▁we ▁are ▁looking ▁for ▁the ▁smallest ▁n ▁such ▁that ▁5 ▁* ▁2 ^ n ▁ ≥ ▁40 .
▁Time ▁for ▁ath le te ▁3 ▁= ▁Total ▁time ▁- ▁( T ime ▁for ▁ath le te ▁1 ▁+ ▁Time ▁for ▁ath le te ▁2 ▁+ ▁Time ▁for ▁ath le te ▁4 )
▁So , ▁the ▁equation ▁based ▁on ▁the ▁amount ▁of ▁silver ▁in ▁the ▁final ▁solution ▁is :
▁Therefore , ▁T ae h y ung ▁has ▁the ▁few est ▁apples .
▁For ▁sh ipp ing ▁purposes , ▁cub es ▁of ▁f ud ge ▁need ▁to ▁be ▁pack aged ▁in ▁boxes ▁that ▁are ▁rect angular ▁pr ism s .
▁Total ▁profit ▁for ▁the ▁month ▁= ▁350 ▁× ▁30
▁The ▁alt itude ▁of ▁a ▁triangle ▁is ▁per pend ic ular ▁to ▁the ▁base ' ; ▁from ▁which ▁we ▁conclud e ▁that ▁the ▁triangles ▁D elt a ▁C A D ▁and ▁D elt a ▁C B D
▁The ▁next ▁term ▁of ▁the ▁root ▁is ▁found ▁by ▁div iding ▁− 3 x 5 ▁, ▁the ▁first ▁term ▁of ▁the ▁remainder ▁after ▁x 6 ▁is ▁sub tra cted , ▁by ▁3 x 4 ▁, ▁and ▁the ▁first ▁complete ▁divis or , ▁3 x 4 ▁− ▁3 x 3 ▁+ ▁x 2 ▁, ▁is ▁found ▁by ▁an nex ing ▁to ▁the ▁trial ▁divis or ▁(3 x 2 ▁− ▁x ) ( − x ), ▁which ▁expression ▁correspond s ▁to ▁(3 a ▁+ ▁b ) b ▁in ▁the ▁typ ical ▁form .
▁So , ▁if ▁we ▁start ▁again ▁with ▁the ▁two ▁triangles ▁ABC ▁and ▁A ' B ' C ' with ▁correspond ing ▁angles ▁of ▁the ▁same ▁measure , ▁then ▁we ▁should ▁first ▁ask : ▁is ▁the ▁o ri ent ation ▁A - > ▁B - > ▁C ▁the ▁same ▁as ▁the ▁o ri ent ation ▁A ' - > ▁B ' - > ▁C ' ▁( b oth ▁cl ock wise ▁or ▁both ▁counter ▁cl ock wise ) ?
▁Since ▁the ▁square ▁of ▁a ▁+ ▁b ▁is ▁a 2 ▁+ ▁2 ab ▁+ ▁b 2 ▁, ▁the ▁square ▁root ▁of ▁a 2 ▁+ ▁2 ab ▁+ ▁b 2 ▁is ▁a ▁+ ▁b .
▁She ▁bought ▁10 ▁more ▁yellow ▁but t ons ▁than ▁the ▁green ▁but t ons , ▁so ▁she ▁bought ▁90 ▁+ ▁10 ▁= ▁100 ▁yellow ▁but t ons .
▁The ▁amount ▁of ▁sand ▁needed ▁for ▁an ▁area ▁that ▁is ▁1.5 ▁times ▁as ▁long ▁and ▁1.5 ▁times ▁as ▁wide ▁would ▁be ▁0 .25 x ( 1 .5 7 9) x ( 1 .5 x ▁12 ) = 60 .75 ▁cub ic ▁feet .
▁To ▁solve ▁this ▁problem , ▁we ▁need ▁to ▁first ▁determine ▁the ▁st o ich i omet ry ▁of ▁the ▁re act ions ▁and ▁then ▁calculate ▁the ▁amount s ▁of ▁re act ants ▁required .
▁In ▁the ▁next ▁chapter ▁we ' ll ▁see ▁this ▁follows ▁from ▁the ▁Py th ag ore an ▁the ore m ; ▁but ▁it ▁is ▁also ▁a ▁fact ▁of ▁E uc l ide an ▁geomet ry , ▁as ▁will ▁be ▁shown ▁in ▁the ▁app end ix .
▁Qu ant ity ▁of ▁mango es ▁= ▁Total ▁amount ▁spent ▁on ▁mango es ▁/ ▁R ate ▁per ▁kg ▁of ▁mango es
▁20 ▁inches , ▁or ▁a ▁little ▁more ▁that ▁20 ▁inches .
▁So , ▁at ▁the ▁start ▁of ▁the ▁fifth ▁month , ▁he ▁we ighed :
▁We ▁have ▁stopped ▁the ▁comput ation ▁at ▁the ▁length ▁12 ▁units , ▁for ▁from ▁that ▁point ▁on ▁this ▁is ▁not ▁the ▁length ▁of ▁the ▁" l ong ▁side ."
▁If ▁we ▁multiply ▁the ▁left ▁sides ▁together ▁we ▁get ▁ABC , ▁and ▁if ▁we ▁multiply ▁the ▁right ▁sides ▁together ▁we ▁get ▁a ^{2} b ^{2} c ^{2} = V ^{2} , ▁since ▁the ▁volume ▁V = ab c .
▁In ▁pack aging , ▁one ▁must ▁know ▁the ▁ed ge ▁d imens ions , ▁the ▁surface ▁area ▁and ▁the ▁volume ; ▁however , ▁some ▁of ▁these ▁are ▁more ▁significant ▁that ▁others .
▁If ▁correspond ing ▁angles ▁of ▁triangles ▁ABC ▁and ▁A ' B ' C ' ▁have ▁the ▁same ▁measure , ▁the ▁triangles ▁are ▁similar .
▁Let ' s ▁assume ▁the ▁cost ▁price ▁of ▁each ▁article ▁is ▁$ C .
▁- ▁P ▁is ▁the ▁principal ▁amount ▁( the ▁sum ▁l ent )
▁2. ▁Des crib e ▁the ▁different ▁2 D ▁cross - se ctions ▁that ▁will ▁result ▁depending ▁on ▁how ▁you ▁sl ice ▁the ▁3 D ▁figure .
▁If ▁Sub ash ▁and ▁P ra k ash ▁together ▁can ▁copy ▁300 ▁pages ▁in ▁40 ▁hours , ▁their ▁combined ▁rate ▁should ▁be :
▁First , ▁we ▁should ▁suspect ▁that ▁the ▁answer ▁is ▁" y es " ▁since ▁it ▁takes ▁3 ▁numbers ▁( the ▁lengths ▁of ▁the ▁sides ) ▁to ▁determine ▁the ▁volume , ▁and ▁if ▁we ▁are ▁given ▁the ▁areas ▁of ▁the ▁faces , ▁we ▁have ▁3 ▁numbers , ▁so ▁that ▁should ▁suff ice .
▁To ▁find ▁the ▁quot ient ▁of ▁the ▁larger ▁number ▁divided ▁by ▁the ▁smaller ▁number , ▁we ▁first ▁identify ▁the ▁larger ▁and ▁smaller ▁numbers .
▁Bet ty ▁e ats ▁1 ▁bro wn ie ▁a ▁day , ▁so ▁in ▁a ▁week , ▁she ▁will ▁eat ▁1 ▁bro wn ie / day ▁* ▁7 ▁days ▁= ▁7 ▁bro wn ies .
▁Now , ▁let ' s ▁convert ▁the ▁percent ages ▁to ▁dec im als ▁and ▁write ▁the ▁equation :
▁So , ▁the ▁number ▁of ▁ch airs ▁left ▁to ▁move ▁is ▁74 ▁ch airs ▁( total ) ▁- ▁57 ▁ch airs ▁( m o ved ) ▁= ▁17 ▁ch airs .
▁The ▁situation ▁is ▁visual ized ▁in ▁Figure ▁15 .
▁The ▁formula ▁for ▁the ▁future ▁value ▁with ▁comp ound ▁interest ▁is :
▁Prof it ▁or ▁L oss ▁Per cent ▁= ▁-1 .6 84 % ▁( app ro xim ately )
▁If ▁k ▁= ▁2, ▁then ▁15 ^ k ▁= ▁15 ^2 ▁= ▁22 5, ▁and ▁22 5 ▁is ▁a ▁divis or ▁of ▁any ▁number ▁that ▁is ▁a ▁multiple ▁of ▁22 5.
▁This ▁means ▁there ▁are ▁33 ▁multip les ▁of ▁6 ▁up ▁to ▁200 , ▁which ▁confirm s ▁that ▁the ▁other ▁number ▁is ▁indeed ▁3.
▁A th le te ▁3 ▁ran ▁15 ▁seconds ▁less ▁than ▁ath le te ▁2.
▁Att ach ▁all ▁points ▁on ▁A ▁to F ▁by ▁line ▁se gments .
▁For ▁two ▁similar ▁triangles , ▁the ▁rati os ▁of ▁correspond ing ▁sides ▁are ▁all ▁the ▁same .
▁When ▁the ▁new ▁person ▁comes ▁in , ▁repl ac ing ▁the ▁one ▁who ▁we igh s ▁75 ▁kg , ▁the ▁total ▁weight ▁becomes :
▁Since ▁the ▁angles ▁C A B ▁and ▁C ' A ' B ' ▁have ▁the ▁same ▁measure , ▁the ▁rot ation ▁must ▁move ▁line ▁segment ▁AB ▁so ▁that ▁it ▁lies ▁on ▁A ' B '.
▁Since ▁Pr inter ▁A ' s ▁rate ▁is ▁10 /3 ▁pages ▁per ▁minute , ▁it ▁would ▁take ▁Pr inter ▁A :
▁Paul ▁was ▁25 ▁minutes ▁late ▁to ▁the ▁party .
▁We ▁now ▁focus ▁on ▁two - d imens ional ▁aspects ▁of ▁sol id ▁sh ap es : ▁plane ▁sections .
▁Now , ▁pl ug ▁in ▁the ▁values ▁we ▁know :
▁- ▁80% ▁of ▁flu ore s cent ▁b ul bs ▁are ▁sw it ched ▁on , ▁which ▁is ▁0.8 0 F .
▁Number ▁of ▁cakes ▁bought ▁= ▁Init ial ▁number ▁of ▁cakes ▁- ▁Number ▁of ▁cakes ▁left
▁Su z an ne ▁and ▁her ▁husband ▁each ▁have ▁1 ▁whole ▁sl ice , ▁so ▁that ' s ▁2 ▁slic es ▁for ▁them .
▁Thus , ▁in ▁Figure ▁1, ▁the ▁angles ▁at ▁V ▁with ▁arrows ▁are ▁vertical ▁angles , ▁as ▁is ▁the ▁pair ▁at ▁V ▁without ▁arrows .
▁This ▁means ▁that ▁Roh an ▁spe nds ▁90% ▁of ▁his ▁salary ▁and ▁sa ves ▁the ▁remaining ▁10% .
▁Rem aining ▁work ▁= ▁1200 ▁work ▁units ▁- ▁4 20 ▁work ▁units
▁Techn ically , ▁these ▁should ▁be ▁called ▁right ▁pr ism s , ▁with ▁right ▁spec ifying ▁that ▁these ▁figures ▁have ▁been ▁drawn ▁out ▁in ▁a ▁direction ▁per pend ic ular ▁to ▁the ▁plane ▁of ▁the ▁start ▁figure .
▁Total ▁eth an ol ▁= ▁E th an ol ▁in ▁gas oh ol ▁+ ▁Add ed ▁eth an ol
▁The ▁traditional ▁argument ▁( and ▁that ▁which ▁appears ▁in ▁grade ▁7 ) ▁is ▁this : ▁both ▁angles ▁AVB ▁and ▁C VD ▁are ▁supplementary ▁to ▁B V C ▁( rec all ▁that ▁two ▁angles ▁are ▁supplementary ▁if ▁they ▁add ▁to ▁a ▁straight ▁angle ), ▁and ▁therefore ▁must ▁have ▁equal ▁measure .
▁So , ▁Jan ine ' s ▁c ater ing ▁service ▁needs ▁to ▁make ▁a ▁total ▁of ▁750 ▁min i ▁me at ▁p ies ▁for ▁the ▁may or ' s ▁birthday .
▁An ▁angle ▁and ▁its ▁image ▁have ▁the ▁same ▁measure .
▁There ▁are ▁4 ▁p airs ▁that ▁sum ▁to ▁9.
▁Now ▁the ▁Py th ag ore an ▁the ore m ▁describ es ▁a ▁relation ▁among ▁the ▁lengths ▁of ▁the ▁sides ▁of ▁a ▁right ▁triangle ; ▁it ▁is ▁also ▁true ▁that ▁this ▁relation ▁describ es ▁a ▁right ▁triangle .
▁Now , ▁we ▁know ▁that ▁1/ 2 ▁of ▁the ▁Dutch ▁were ▁also ▁American .
▁We ▁start ▁this ▁section ▁by ▁turning ▁this ▁suggest ion ▁into ▁an ▁" ex pl anation ▁of ▁a ▁pro of ," ▁and ▁we ▁give ▁one ▁other ▁way ▁of ▁seeing ▁that ▁this ▁is ▁true .
▁Let ' s ▁complete ▁by ▁showing ▁that ▁an ▁angle ▁and ▁its ▁image ▁under ▁a ▁d ilation ▁are ▁actually ▁con g ru ent .
▁a ▁par alle lo gram ▁that ▁is ▁not ▁a ▁rectangle
▁1. ▁Sam antha ' s ▁last ▁name ▁has ▁three ▁fewer ▁letters ▁than ▁Bob b ie ' s ▁last ▁name :
▁Now ▁what ▁volume ▁box ▁do ▁I ▁need ?
▁Per cent age ▁of ▁eth an ol ▁= ▁( T otal ▁eth an ol ▁/ ▁Total ▁volume ) ▁* ▁100
▁So , ▁his ▁initial ▁weight ▁was :
▁Let ' s ▁find ▁a ▁common ▁factor ▁for ▁the ▁fl av oring ▁in ▁both ▁rati os ▁to ▁comb ine ▁them ▁into ▁a ▁single ▁ratio .
▁A ▁plane ▁section ▁of ▁a ▁sol id ▁in ▁3 D ▁is ▁the ▁2 ▁d imens ional ▁figure ▁one ▁gets ▁by ▁slic ing ▁the ▁sol id ▁along ▁some ▁plane ▁in ▁space .
▁S av ings ▁per ▁b age l ▁= ▁ € 0 .3 3 3 3 ▁( rounded ▁to ▁four ▁decimal ▁places )
▁What ▁is ▁the ▁volume ▁of ▁the ▁box ▁I ▁would ▁need ?
▁In ▁any ▁problem , ▁pick ▁a ▁face ▁to ▁be ▁the ▁base ▁if ▁it ▁makes ▁the ▁problem ▁simpl er ; ▁other wise ▁it ▁doesn ' t ▁matter .
▁The ▁combined ▁total ▁score ▁for ▁both ▁teams ▁is ▁62 ▁points .
▁50 ▁pages ▁+ ▁75 ▁pages ▁+ ▁60 ▁pages ▁+ ▁100 ▁pages ▁= ▁28 5 ▁pages
▁Now , ▁let ' s ▁compare ▁the ▁total ▁number ▁of ▁blue ▁and ▁red ▁ball oons ▁with ▁the ▁total ▁number ▁of ▁yellow ▁and ▁green ▁ball oons :
▁What ▁we ▁have ▁to ▁find , ▁given ▁the ▁two ▁triangles , ▁is ▁a ▁sequ ence ▁of ▁rigid ▁mot ions ▁that ▁takes ▁one ▁on to ▁the ▁other .
▁Therefore , ▁the ▁delivery ▁dri ver ▁has ▁to ▁make ▁3 ▁more ▁st ops .
▁Jacob ▁will ▁be ▁8 ▁years ▁old ▁in ▁4 ▁years .
▁Se an ▁and ▁Ind ira ▁played ▁cricket ▁together ▁for ▁a ▁total ▁of ▁15 12 ▁minutes .
▁The ▁7 th ▁gr ad ers ▁at ▁Albert ▁R . ▁L y man ▁Middle ▁School ▁were ▁helping ▁to ▁re n o vate ▁a ▁play ground ▁for ▁the ▁kind erg art ners ▁at ▁the ▁near by ▁B land ing ▁E lementary ▁School .
▁Now ▁rot ate ▁triangle ▁I ▁( in ▁its ▁new ▁position ) ▁by ▁90 degrees .
▁However , ▁we ▁can ▁try ▁to ▁ded uce ▁a ▁possible ▁pattern ▁by ▁making ▁some ▁assum pt ions .
▁Thus ▁all ▁length ▁change ▁by ▁the ▁factor ▁2, ▁and ▁area ▁changes ▁by ▁the ▁factor ▁2 ^{2} = 4.
▁The ▁second ▁fraction ▁is ▁already ▁in ▁24 th s , ▁so ▁we ▁can ▁subtract ▁directly :
▁The ▁resulting ▁3 D ▁figure ▁is ▁called ▁a ▁pr ism .
▁The ▁Greeks ▁went ▁further ▁with ▁a ▁specific ▁py ram id , ▁the ▁t et ra hed ron .
▁If ▁the ▁sections ▁are ▁all ▁of ▁the ▁same ▁size ▁and ▁shape , ▁we ▁have ▁a ▁pr ism ▁and ▁if ▁the ▁they ▁are ▁sc al ings ▁of ▁the ▁original ▁we ▁have ▁a ▁con e .
▁Let ' s ▁assume ▁the ▁total ▁work ▁is ▁represented ▁by ▁the ▁number ▁1 ▁( as ▁a ▁whole ), ▁and ▁K ir an ▁completed ▁1/3 ▁of ▁it ▁in ▁some ▁days .
▁Let ' s ▁denote ▁the ▁percentage ▁of ▁silver ▁in ▁the ▁first ▁solution ▁as ▁x %.
▁What ▁is ▁the ▁length ▁of ▁the ▁long est ▁line ▁segment ▁in ▁a ▁box ▁of ▁wid th ▁10 ", ▁length ▁16 " ▁and ▁height ▁8 " ?
▁Total ▁profit ▁for ▁the ▁month ▁= ▁Me an ▁daily ▁profit ▁× ▁Number ▁of ▁days
▁In ▁Figure ▁16 , ▁we ▁want ▁to ▁find ▁the ▁length ▁of ▁the ▁d as hed ▁line ▁from ▁A ▁to ▁B .
▁Now , ▁we ▁divide ▁the ▁total ▁minutes ▁by ▁the ▁length ▁of ▁each ▁ep is ode ▁to ▁find ▁the ▁number ▁of ▁ep is od es :
▁Therefore , ▁21 % ▁of ▁the ▁population ▁in ▁town ▁P ▁are ▁employ ed ▁mal es .
▁For ▁example , ▁the ▁number ▁of ▁exp osed ▁faces ▁of ▁each ▁unit ▁cub e ▁is ▁different ▁for ▁each ▁pr ism .
▁The ▁tr ig on omet r ical ▁form ul æ , ▁given ▁above , ▁are ▁ex amples ▁of ▁this ▁fact .
▁10 8 ▁degrees ▁/ ▁3 60 ▁degrees ▁= ▁Man ufact uring ▁employees ▁percentage ▁/ ▁100%
▁Now , ▁because ▁of ▁(1) ▁above , ▁that ▁opp osing ▁angles ▁at ▁a ▁ver te x ▁are ▁of ▁equal ▁measure , ▁we ▁can ▁conclud e ▁that ▁the ▁angle ▁den ot ed ▁by ▁the ▁d as hed ▁ar c ▁is ▁also ▁equal ▁to ▁the ▁angles ▁den ot ed ▁by ▁the ▁sol id ▁ar c .
▁If ▁all ▁students ▁in ▁the ▁class ▁get ▁this ▁close , ▁all ▁with ▁different ▁figures , ▁then ▁that ▁is ▁substant ial ▁stat istical ▁evidence ▁for ▁the ▁claim ▁that ▁the ▁triangle ▁is ▁always ▁a ▁right ▁triangle .
▁The ▁total ▁runs ▁scored ▁after ▁11 ▁innings ▁is ▁11 ▁times ▁the ▁new ▁average , ▁which ▁is ▁11 ▁* ▁5 9 ▁= ▁64 9.
▁Then , ▁any ▁triangle ▁with ▁two ▁side ▁lengths ▁a ▁and ▁b ▁is ▁con g ru ent ▁to ▁a ▁triangle ▁with ▁one ▁side ▁BC , ▁and ▁the ▁other ▁side ▁the ▁line ▁segment ▁from ▁C ▁to ▁a ▁point ▁A ▁on ▁the ▁circle .
▁Despite ▁this ▁special ▁deal , ▁the ▁deal er ▁still ▁makes ▁a ▁profit ▁of ▁36 %.
▁By ▁the ▁Py th ag ore an ▁the ore m , ▁the ▁length ▁of ▁P Q ▁is ▁s q r t { 4 ^{2} +3 ^{2} } = 5.
▁D il an ▁was ▁16 5 ▁- ▁10 ▁= ▁15 5 ▁minutes ▁late ▁to ▁the ▁party .
▁Since ▁his ▁basket ▁can ▁hold ▁12 ▁apples ▁when ▁full , ▁the ▁number ▁of ▁more ▁apples ▁Jack ' s ▁basket ▁can ▁currently ▁hold ▁is :
▁If ▁we ▁f old ▁it ▁three ▁times , ▁we ▁get ▁8 ▁smaller ▁pieces , ▁and ▁so ▁on .
▁B ren ' s ▁age ▁10 ▁years ▁from ▁now ▁= ▁40 ▁+ ▁10 ▁= ▁50 ▁years
▁To ▁find ▁out ▁how ▁many ▁books ▁she ▁sold , ▁we ▁subtract ▁the ▁number ▁of ▁books ▁she ▁has ▁left ▁from ▁the ▁original ▁number ▁of ▁books ▁she ▁had :
▁Now ▁the ▁area ▁of ▁the ▁base ▁is ▁150 x ▁200 = 30 ,000 ▁s q .
▁Cong ru ent ▁figures ▁are ▁similar .
▁Let ' s ▁denote ▁the ▁number ▁of ▁pens ▁J ane ▁has ▁as ▁J .
▁That ▁is ▁enough ; ▁apply ▁the ▁d ilation ▁with ▁center ▁at ▁A ▁and ▁factor ▁r = | A B | / | A B | .
▁Of ▁the ▁three ▁pr ism s ▁illust rated ▁in ▁Figure ▁13, ▁the ▁first ▁two ▁can ▁be ▁view ed ▁as ▁having ▁been ▁formed ▁by ▁drawing ▁out ▁any ▁one ▁of ▁its ▁faces ▁in ▁the ▁per pend ic ular ▁direction .
▁Finally , ▁we ▁add ▁the ▁regular ▁season ▁games ▁to ▁the ▁additional ▁ri val ry ▁games ▁to ▁get ▁the ▁total ▁number ▁of ▁games ▁played :
▁She ▁gives ▁each ▁friend ▁1/ 4 ▁of ▁her ▁stra w ber ries , ▁so ▁each ▁friend ▁gets :
▁Therefore , ▁the ▁desired ▁percentage ▁of ▁eth an ol ▁in ▁the ▁mixture ▁for ▁opt im um ▁performance ▁is ▁10% .
▁M iles ▁and ▁Kat y ' s ▁combined ▁flour ▁= ▁14 1 .0 96 ▁o un ces ▁+ ▁240 ▁o un ces ▁= ▁3 81 .0 96 ▁o un ces
▁As ▁before , ▁here ▁the ▁objective ▁is ▁to ▁give ▁students ▁an ▁inform al ▁and ▁int u itive ▁understanding ▁of ▁these ▁facts ▁about ▁angles ▁and ▁triangles ; ▁all ▁this ▁material ▁will ▁be ▁res um ed ▁in ▁Second ary ▁Mat hem at ics ▁in ▁a ▁more ▁formal ▁and ▁log ically ▁consist ent ▁exp osition .
▁The ▁total ▁produ ces ▁21 6 +2 x ▁450 + 5 40 = 16 5 6 ▁s q .
▁Per cent age ▁Inc re ase ▁= ▁[ ( F inal ▁Val ue ▁- ▁Init ial ▁Val ue ) ▁/ ▁Init ial ▁Val ue ] ▁* ▁100
▁Since ▁we ▁need ▁3 ▁m oles ▁of ▁Ca ( OH ) 2, ▁we ▁will ▁need ▁2 ▁m oles ▁of ▁H 3 P O 4 ▁to ▁re act ▁completely .
▁A ▁pr ism ▁is ▁described ▁as ▁the ▁sol id ▁formed ▁by ▁drawing ▁out ▁a ▁figure ▁on ▁a ▁plane ▁for ▁a ▁spec ified ▁distance ▁along ▁par alle l ▁lines ▁e man ating ▁from ▁the ▁plane .
▁Then ▁the ▁area ▁of ▁the ▁original ▁figure ▁is ▁the ▁sum ▁of ▁the ▁areas ▁of ▁the ▁component ▁figures .
▁Not ice ▁that ▁if ▁any ▁two ▁selected ▁points ▁are ▁equ id istant ▁from ▁the ▁original ▁ver te x , ▁the ▁cross ▁section ▁would ▁be ▁an ▁is os ce les ▁triangle .
▁But ▁the ▁par alle l ism ▁between ▁geomet ry ▁and ▁algebra ▁can ▁be ▁p ushed ▁still ▁further , ▁o w ing ▁to ▁the ▁fact ▁that ▁lengths , ▁areas , ▁vol um es , ▁and ▁angles ▁are ▁all ▁meas ur able ; ▁so ▁that , ▁for ▁example , ▁the ▁size ▁of ▁any ▁length ▁can ▁be ▁determined ▁by ▁the ▁number ▁( not ▁necess arily ▁integral ) ▁of ▁times ▁which ▁it ▁contains ▁some ▁ar bit r arily ▁known ▁unit , ▁and ▁similar ly ▁for ▁areas , ▁vol um es , ▁and ▁angles .
▁The ▁fin er ▁the ▁gr id ▁the ▁better ▁the ▁approxim ation .
▁E th an ol ▁in ▁gas oh ol ▁= ▁5% ▁of ▁45 ▁liters
▁2 ▁pack ages ▁* ▁12 ▁te es / p ack age ▁= ▁24 ▁te es
▁10 ▁she ets ▁of ▁paper ▁* ▁3 ▁stick ers ▁per ▁she et ▁= ▁30 ▁stick ers
▁Now , ▁transl ate ▁V ▁to ▁V '.
▁So , ▁the ▁top ▁of ▁the ▁lad der ▁h its ▁the ▁wall ▁about ▁16 ▁feet ▁above ▁the ▁ground .
▁Total ▁eth an ol ▁= ▁2 .25 ▁liters ▁+ ▁2.5 ▁liters
▁5. ▁P ip er ▁div ides ▁the ▁remaining ▁b ars ▁among ▁herself ▁and ▁her ▁5 ▁friends ▁(6 ▁people ▁in ▁total ):
▁Now , ▁let ' s ▁find ▁out ▁how ▁many ▁f ruits ▁Kar le e ▁is ▁left ▁with :
▁Since ▁it ▁is ▁formed ▁by ▁swe eping ▁out ▁a ▁circle ▁and ▁not ▁a ▁poly gon , ▁it ▁is ▁not ▁a ▁pr ism .
▁So , ▁the ▁large ▁bot tle ▁of ▁the ▁s port ▁form ulation ▁contains ▁15 ▁o un ces ▁of ▁corn ▁sy ru p .
▁First , ▁let ' s ▁find ▁out ▁how ▁many ▁people ▁on ▁the ▁bus ▁were ▁Dutch .
▁- ▁Paul ▁was ▁25 ▁minutes ▁late .
▁When ▁the ▁person ▁we igh ing ▁92 ▁kg ▁leaves ▁and ▁a ▁new ▁person ▁comes ▁in , ▁the ▁average ▁weight ▁increases ▁by ▁4.5 ▁kg .
▁Not ice ▁that ▁surface ▁area ▁decre ases ▁as ▁the ▁rect angular ▁pr ism s ▁move ▁cl oser ▁to ▁the ▁shape ▁of ▁the ▁cub e .
▁The ▁fraction ▁7 / 8 ▁is ▁in ▁its ▁simpl est ▁form , ▁so ▁we ▁can ▁find ▁equivalent ▁fractions ▁by ▁multiply ing ▁both ▁the ▁numer ator ▁( 7) ▁and ▁the ▁denominator ▁( 8) ▁by ▁the ▁same ▁integer .
▁The ▁length ▁of ▁the ▁horizontal ▁leg ▁is ▁7 - ( - 3) ▁= ▁10, ▁and ▁that ▁of ▁the ▁vertical ▁leg ▁is ▁7 -2 ▁= ▁5.
▁B ars ▁for ▁each ▁person ▁without ▁extra ▁= ▁300 ▁/ ▁9 ▁= ▁33 .3 3 ▁b ars
▁After ▁500 ▁people ▁took ▁1 ▁can ▁each , ▁she ▁was ▁left ▁with :
▁The ▁first ▁has ▁the ▁advant age ▁that ▁it ▁us es ▁a ▁construction ▁with ▁which ▁the ▁student ▁is ▁famil iar ▁( that ▁to ▁find ▁the ▁area ▁of ▁a ▁triangle ) ▁and ▁thus ▁reinfor ces ▁that ▁idea .
▁B ars ▁for ▁each ▁person ▁= ▁19 2 ▁/ ▁6 ▁= ▁32 ▁b ars
▁Move ▁the ▁segment ▁AB , ▁by ▁rigid ▁mot ions ▁as ▁above , ▁so ▁that A ▁fall s ▁on ▁C , ▁and ▁AB ▁and ▁C D ▁now ▁lie ▁on ▁the ▁same ▁ray ▁starting ▁from ▁C .
▁A ▁b aker ▁creat es ▁f ant astic ▁cup c akes ▁that ▁can ▁each ▁be ▁comfort ably ▁enclosed ▁in ▁a ▁cub e ▁of ▁side ▁length ▁2 ▁in ▁and ▁square ▁base .
▁Total ▁percentage ▁of ▁Muslims , ▁Hind us , ▁and ▁S ik h s :
▁Kel ly ▁bought ▁approximately ▁1.3 15 5 ▁p ounds ▁of ▁sn ack s ▁in ▁total .
▁So ▁the ▁length ▁of ▁the ▁rect angular ▁garden ▁is ▁150 ▁meters .
▁First ▁of ▁all , ▁to ▁make ▁the ▁argument ▁more ▁clear : ▁let ' s ▁design ate ▁the ▁sides ▁of ▁ABC ▁by ▁the ▁lower ▁case ▁version ▁of ▁the ▁lab el ▁of ▁the ▁opposite ▁ver te x : ▁so , ▁a ▁is ▁the ▁side ▁opposite ▁ver te x ▁A , ▁and ▁so ▁forth .
▁Port ion ▁eaten ▁in ▁4 ▁hours ▁= ▁1/ 5
▁The ▁emp h ases ▁of ▁this ▁section ▁are :
▁S av ings ▁per ▁b age l ▁in ▁c ents ▁= ▁ € 0 .3 3 3 3 ▁* ▁100
▁Now , ▁we ▁find ▁the ▁ratio ▁of ▁C ▁in ▁pa isa ▁to ▁A ▁in ▁Rs . :
▁As ▁the ▁gr id ▁gets ▁fin er ▁and ▁fin er , ▁the ▁approxim ations ▁get ▁better ▁and ▁better , ▁but ▁the ▁formula ▁remains ▁the ▁same .
▁It ▁must ▁be ▁observed ▁that ▁a ▁rep res ents ▁so ▁many ▁tens ▁with ▁respect ▁to ▁the ▁next ▁figure ▁of ▁the ▁root .
▁So , ▁the ▁correct ▁mole c ular ▁weight ▁of ▁water ▁is ▁approximately ▁18 .01 6 ▁g / mol , ▁not ▁7 2.
▁Now , ▁we ▁know ▁that ▁12 % ▁of ▁x ▁plus ▁3 .75 ▁equ als ▁9.1 5 :
▁To ▁find ▁the ▁value ▁of ▁d ▁to ▁the ▁ne arest ▁t enth , ▁we ▁first ▁perform ▁the ▁multip lication ▁and ▁then ▁the ▁division :
▁In ▁an ▁ide al ▁sc en ari o , ▁a ▁b ats man ▁can ▁score ▁the ▁maximum ▁number ▁of ▁runs ▁off ▁every ▁ball ▁bow led ▁to ▁him .
▁So ▁the ▁initial ▁ratio ▁of ▁board ers ▁to ▁day ▁scholars ▁was ▁7 ▁to ▁16 .
▁The ▁first ▁term ▁of ▁the ▁root , ▁x 2 ▁, ▁is ▁obtained ▁by ▁taking ▁the ▁cub e ▁root ▁of ▁the ▁first ▁term ▁of ▁the ▁given ▁expression ; ▁and ▁the ▁first ▁trial d iv is or , ▁3 x 4 ▁, ▁is ▁obtained ▁by ▁taking ▁three ▁times ▁the ▁square ▁of ▁this ▁term .
▁Now , ▁we ▁convert ▁the ▁volume ▁from ▁cub ic ▁feet ▁to ▁gallons ▁using ▁the ▁convers ion ▁factor ▁1 ▁cub ic ▁foot ▁= ▁7 .5 ▁gallons :
▁Since ▁correspond ing ▁lines ▁under ▁a ▁translation ▁are ▁par alle l , ▁the ▁ray ▁V A ▁must ▁go ▁to ▁the ▁ray ▁V ' A ' ▁and ▁the ▁ray ▁V B ▁to ▁the ▁ray ▁V ' B '.
▁Luc ille ▁needs ▁6 ▁can s ▁of ▁blue ▁paint ▁and ▁18 ▁can s ▁of ▁yellow ▁paint .
▁So , ▁the ▁fraction ▁of ▁the ▁selected ▁film s ▁that ▁are ▁in ▁col or ▁is ▁30 /3 1 .
▁For ▁example , ▁the ▁addition ▁of ▁two ▁greater ▁lengths ▁must ▁y ield ▁a ▁length ▁greater ▁than ▁that ▁y ield ed ▁by ▁the ▁addition ▁of ▁two ▁smaller ▁lengths .
▁B ot tle ▁cap s ▁per ▁child ▁= ▁Total ▁bot tle ▁cap s ▁÷ ▁Number ▁of ▁children
▁The ▁language ▁of ▁this ▁standard ▁is ▁very ▁precise : ▁it ▁does ▁not ▁say ▁Pro ve ... ▁but ▁it ▁says ▁Ex pl ain ▁a ▁pro of ▁of ... , ▁suggesting ▁that ▁the ▁point ▁for ▁students ▁is ▁to ▁art ic ulate ▁their ▁understanding ▁of ▁the ▁the ore m ; ▁not ▁to ▁demonst rate ▁sk ill ▁in ▁rec iting ▁a ▁formal ▁pro of .
▁We ek ▁4 : ▁Alex ' s ▁collection ▁dou bles ▁once ▁more , ▁so ▁she ▁has ▁16 ▁* ▁2 ▁= ▁32 ▁pens .
▁The ▁initial ▁purch ase ▁price ▁of ▁the ▁machine ▁was ▁Rs ▁11 000 .
▁Let ' s ▁list ▁out ▁all ▁possible ▁p airs ▁and ▁their ▁sum s :
▁In ▁general ▁a ▁poly gon al ▁figure ▁can ▁be ▁part ition ed ▁into ▁combination ▁of ▁simpl er ▁sh ap es ▁( act ually , ▁triangles ▁and ▁rect angles ) ▁for ▁whose ▁areas ▁form ul as ▁have ▁been ▁developed ▁already .
▁Our ▁figure ▁shows ▁a ▁typ ical ▁such ▁rectangle .
▁This ▁means ▁the ▁selling ▁price ▁is ▁150 % ▁of ▁the ▁total ▁cost ▁including ▁transportation .
▁Therefore , ▁in ▁the ▁first ▁5 ▁days , ▁they ▁all ▁collect ively ▁read ▁28 5 ▁pages .
▁If ▁you ▁have ▁2 ▁m oles ▁of ▁hy dro ch lor ic ▁ac id , ▁you ▁would ▁need ▁2 ▁m oles ▁of ▁s od ium ▁b ic ar bon ate ▁to ▁re act ▁completely ▁with ▁the ▁ac id , ▁according ▁to ▁the ▁st o ich i omet ry ▁of ▁the ▁bal anced ▁equation .
▁A ▁rect angular ▁pr ism ▁has ▁6 ▁faces , ▁ident ical ▁in ▁opp osing ▁p airs .
▁So , ▁the ▁annual ▁interest ▁rate ▁for ▁the ▁loan ▁to ▁Paul ▁is ▁22 %.
▁1 ▁kg ▁is ▁approximately ▁35 .2 7 4 ▁o un ces .
▁Am oura ▁was ▁55 ▁minutes ▁late ▁to ▁the ▁party .
▁So ▁they ▁will ▁need ▁90 ▁bags ▁of ▁sand ▁for ▁the ▁large ▁sw ing ▁set .
▁To ▁find ▁the ▁probability ▁that ▁the ▁sum ▁of ▁two ▁integ ers ▁( one ▁from ▁set ▁a ▁and ▁one ▁from ▁set ▁b ) ▁will ▁equal ▁9, ▁we ▁need ▁to ▁consider ▁all ▁possible ▁p airs ▁and ▁count ▁the ▁number ▁of ▁p airs ▁that ▁sum ▁to ▁9.
▁This ▁may ▁seem ▁unn ecess ary ▁to ▁point ▁out , ▁but ▁it ▁does ▁state ▁a ▁fact : ▁given ▁any ▁points ▁P ▁and ▁Q ▁in ▁the ▁plane , ▁there ▁is ▁a ▁rigid ▁motion ▁taking ▁P ▁to ▁Q .
▁We ▁are ▁given ▁that ▁this ▁new ▁average ▁after ▁the ▁11 th ▁in ning ▁is ▁5 9, ▁so ▁A ▁+ ▁1 ▁= ▁5 9.
▁So , ▁Bill ▁ate ▁half ▁as ▁many ▁p ies ▁as ▁S ier ra , ▁which ▁is ▁12 ▁p ies ▁/ ▁2 ▁= ▁6 ▁p ies .
▁1000 ▁people ▁* ▁2 ▁can s / p erson ▁= ▁2000 ▁can s
▁First ▁transl ate ▁A ▁to ▁the ▁point ▁C , ▁and ▁then ▁rot ate ▁the ▁image ▁of ▁AB ▁so ▁that ▁it ▁and ▁C D ▁lie ▁on ▁the ▁same ▁line .
▁If ▁she ▁has ▁2 ▁times ▁as ▁many ▁or anges ▁as ▁stra w ber ries , ▁then ▁she ▁has :
▁Now , ▁since ▁AC B ▁and ▁A ' C ' B ' ▁have ▁the ▁same ▁measure , ▁the ▁line ▁se gments ▁C B ▁and ▁C ' B ' ▁must ▁be ▁par alle l ▁( by ▁Prop osition ▁2 ).
▁- ▁3 ▁par ake ets , ▁so ▁they ▁eat ▁3 P ▁grams ▁a ▁day ▁together .
▁When ▁you ▁think ▁about ▁a ▁traditional ▁Rub ik ' s ▁C ub e ▁there ▁are ▁three ▁lay ers .
▁3. ▁P ip er ▁takes ▁1/ 4 ▁of ▁the ▁initial ▁number ▁of ▁ch oc olate ▁b ars :
▁12 .75 ▁( n umber ▁of ▁c her ry ▁p ies ▁b ak ed ▁last ▁time ) ▁× ▁5 .7 ▁( m ult ip lier ▁for ▁the ▁bigger ▁event ) ▁= ▁72 .6 75 ▁c her ry ▁p ies
▁Wh ich ▁of ▁the ▁pack ages ▁requ ires ▁the ▁least ▁and ▁the ▁greatest ▁amount ▁of ▁material ▁and ▁why ▁would ▁it ▁be ▁important ?
▁For ▁each ▁power ▁of ▁2, ▁we ▁can ▁comb ine ▁it ▁with ▁any ▁combination ▁of ▁the ▁other ▁prime ▁factors .
▁Now , ▁2 ▁kil ogram s ▁of ▁water ▁ev ap or ate ▁from ▁the ▁solution .
▁So , ▁the ▁least ▁number ▁that ▁should ▁be ▁added ▁to ▁10 5 6 ▁to ▁make ▁it ▁completely ▁divis ible ▁by ▁23 ▁is ▁2.
▁What ▁is ▁the ▁radi us ▁of ▁the ▁se m ic irc le ?
▁The ▁first ▁figure ▁is ▁a ▁representation ▁of ▁a ▁general ▁pr ism .
▁So , ▁if ▁he ▁lost ▁12 ▁p ounds ▁in ▁the ▁fifth ▁month , ▁he ▁must ▁have ▁lost ▁twice ▁that ▁amount ▁in ▁the ▁fourth ▁month :
▁Kar le e ▁gives ▁each ▁friend ▁1/ 5 ▁of ▁her ▁gra p es , ▁so ▁each ▁friend ▁gets :
▁1. ▁Thomas ▁and ▁his ▁8 ▁friends ▁take ▁3 / 8 ▁of ▁the ▁b ars :
▁For ▁a ▁square , ▁the ▁answer ▁is ▁" y es " : ▁if ▁you ▁know ▁the ▁perimeter ▁you ▁can ▁find ▁the ▁side ▁length ▁and ▁from ▁this , ▁the ▁area .
▁Prof it ▁for ▁A ▁= ▁20% ▁of ▁Rs . ▁150
▁We ▁already ▁know ▁Distance _ A ▁is ▁400 ▁km , ▁so ▁we ▁can ▁write :
▁The ▁last ▁is ▁a ▁pr ism , ▁called ▁the ▁tri angular ▁pr ism ▁or ▁w ed ge ▁since ▁it ▁is ▁formed ▁by ▁swe eping ▁out ▁a ▁triangle .
▁To ▁find ▁out ▁what ▁percentage ▁of ▁the ▁fem ales ▁are ▁pregnant , ▁we ▁calculate :
▁B land ing ▁City ▁reg ulations ▁require ▁that ▁the ▁sand ▁under ne ath ▁the ▁sw ings ▁be ▁at ▁least ▁15 ▁inches ▁deep .
▁So ▁the ▁number ▁that ▁has ▁the ▁same ▁ratio ▁to ▁4 ▁minutes ▁as ▁12 ▁seconds ▁to ▁8 ▁seconds ▁is ▁3 60 ▁seconds .
▁To ▁find ▁the ▁number ▁of ▁boys ▁who ▁belong ▁to ▁other ▁communities , ▁we ▁calculate ▁22 % ▁of ▁the ▁total ▁number ▁of ▁boys ▁in ▁the ▁school :
▁It ▁will ▁be ▁not iced ▁that ▁each ▁success ive ▁trial - d iv is or ▁may ▁be ▁obtained ▁by ▁taking ▁the ▁preced ing ▁complete ▁divis or ▁with ▁its ▁last ▁term ▁dou bled .
▁Total ▁mole c ular ▁weight ▁of ▁the ▁comp ound ▁= ▁200 ▁am u
▁Equ ation ▁1 ▁( T otal ▁cost ▁including ▁tax ):
▁4 ▁members ▁* ▁20 ▁te es / m ember ▁= ▁80 ▁te es
▁We ▁conclud e ▁that ▁the ▁d as hed ▁lines ▁are ▁all ▁of ▁the ▁same ▁length , ▁tell ing ▁us ▁that ▁the ▁d as hed ▁figure ▁is ▁a ▁r h omb us .
▁Are a ▁= ▁W id th ▁x ▁He ight ▁= ▁3 m ▁x ▁2 m ▁= ▁6 m 2
▁Since ▁each ▁bird ▁we igh s ▁3 ▁l bs , ▁no ▁more ▁birds ▁can ▁join ▁without ▁the ▁f ence ▁bre aking ▁because ▁even ▁one ▁more ▁bird ▁would ▁add ▁3 ▁l bs , ▁which ▁is ▁more ▁than ▁the ▁remaining ▁2 ▁l bs ▁capacity ▁of ▁the ▁f ence .
▁Work ▁done ▁by ▁An an th u ▁= ▁72 B ▁= ▁72 ( W ▁/ ▁D )
▁99 ▁lo aves ▁/ ▁33 ▁ban anas ▁= ▁3 ▁lo aves ▁per ▁ban ana
▁That ▁is , ▁we ▁will ▁tra ce ▁everything ▁ult im ately ▁back ▁to ▁the ▁area ▁of ▁squ ares , ▁because ▁we ▁measure ▁area ▁in ▁square ▁units .
▁The ▁effect ▁is ▁to ▁lose ▁some ▁area ▁at ▁the ▁short ▁end ▁but ▁gain ▁more ▁area ▁on ▁the ▁long ▁side ▁of ▁the ▁rectangle ▁for ▁an ▁overall ▁increase ▁in ▁area .
▁These ▁ex amples ▁show ▁that ▁we ▁can ▁use ▁the ▁Py th ag ore an ▁the ore m ▁to ▁find ▁lengths ▁of ▁line ▁se gments ▁in ▁space .
▁200 ▁grams ▁= ▁200 / 45 4 ▁p ounds ▁( since ▁there ▁are ▁approximately ▁45 4 ▁grams ▁in ▁a ▁p ound )
▁This ▁would ▁produce ▁2 ▁m oles ▁of ▁water , ▁2 ▁m oles ▁of ▁car bon ▁d io x ide , ▁and ▁2 ▁m oles ▁of ▁s od ium ▁ch lor ide , ▁satisf ying ▁the ▁condition ▁that ▁the ▁number ▁of ▁m oles ▁of ▁each ▁product ▁is ▁equal .
▁Total ▁blue ▁and ▁red ▁ball oons ▁= ▁104 ▁+ ▁26
▁Let ' s ▁denote ▁Han by ul ' s ▁weight ▁as ▁H b ▁and ▁H ans ol ' s ▁weight ▁as ▁H s .
▁So , ▁there ▁are ▁21 ▁employ ed ▁mal es ▁in ▁town ▁P .
▁Let ' s ▁expl ore ▁the ▁first ▁two ▁poss ib ilities :
▁Then ▁the ▁d ilation ▁of ▁factor ▁r ▁places ▁the ▁first ▁circle ▁on ▁top ▁of ▁the ▁other .
▁Make ▁a ▁box ▁of ▁the ▁d imens ions ▁of ▁circum sc rib ed ▁cub e , ▁and ▁a ▁py ram id ▁as ▁indicated ▁by ▁the ▁figure .
▁Now ▁we ▁want ▁to ▁find ▁the ▁percentage ▁above ▁the ▁cost ▁price ▁at ▁which ▁the ▁marked ▁price ▁is ▁fixed .
▁To ▁calculate ▁the ▁trad er ' s ▁overall ▁profit ▁or ▁loss ▁percent , ▁we ▁need ▁to ▁first ▁determine ▁the ▁cost ▁price ▁of ▁each ▁car ▁and ▁then ▁calculate ▁the ▁total ▁cost ▁price ▁and ▁total ▁selling ▁price .
▁Section ▁2 ▁is ▁about ▁standard ▁8 G 6 : ▁Ex pl ain ▁a ▁pro of ▁of ▁the ▁Py th ag ore an ▁The ore m ▁and ▁its ▁con verse .
▁+ ▁B ars ▁returned ▁by ▁C ather ine ▁= ▁15
▁Then ▁the ▁length ▁of ▁the ▁dia gon al ▁on ▁the ▁bott om ▁face ▁is ▁s q r t {1 ^{2} +1 ^{2} } = s q r t { 2} ▁units , ▁and ▁the ▁length ▁of ▁the ▁dia gon al ▁AB ▁is ▁s q r t {1 ^{2} + ( s q r t { 2} ) ^{2} } = s q r t { 3 } .
▁This ▁expression ▁may ▁be ▁obtained ▁by ▁adding ▁to ▁the ▁preced ing ▁complete ▁divis or , ▁3 a 2 ▁+ ▁3 ab ▁+ ▁b 2 ▁, ▁its ▁second ▁term ▁and ▁twice ▁its ▁third ▁term .
▁3. ▁She ▁find s ▁another ▁45 ▁se as he lls , ▁so ▁she ▁now ▁has ▁16 ▁+ ▁45 ▁= ▁61 ▁se as he lls .
▁What ▁sh ap es ▁can ▁be ▁created ▁by ▁one ▁sl ice ▁through ▁a ▁circular ▁cyl inder ?
▁First ▁we ▁dec om p ose ▁the ▁area ▁into ▁sh ap es ▁that ▁we ▁know ▁well , ▁a ▁se mi - c irc le ▁and ▁a ▁rectangle .
▁Let ' s ▁convert ▁the ▁f rog ' s ▁j ump ▁from ▁meters ▁to ▁inches , ▁and ▁then ▁we ▁can ▁add ▁the ▁two ▁j um ps ▁together ▁and ▁convert ▁the ▁total ▁to ▁feet .
▁Let ' s ▁denote ▁the ▁probability ▁of ▁getting ▁the ▁computer ▁hard w are ▁contract ▁as ▁P ( H ) ▁and ▁the ▁probability ▁of ▁getting ▁the ▁soft w are ▁contract ▁as ▁P ( S ).
▁The ▁second ▁term ▁of ▁the ▁root , ▁− 2 x 2 ▁y , ▁is ▁obtained ▁by ▁div iding ▁− 20 x 3 ▁y ▁by ▁10 x , ▁the ▁double ▁of ▁5 x , ▁and ▁this ▁new ▁term ▁of ▁the ▁root ▁is ▁also ▁an nex ed ▁to ▁the ▁divis or , ▁10 x , ▁to ▁complete ▁the ▁divis or .
▁The ▁situation ▁is ▁depic ted ▁in ▁Figure ▁17.
▁Students ▁taking ▁only ▁music ▁= ▁Total ▁students ▁taking ▁music ▁- ▁Students ▁taking ▁both ▁music ▁and ▁art
▁How ▁do ▁we ▁figure ▁out ▁the ▁area ▁of ▁a ▁poly gon al ▁shape ?
▁3. ▁Y o ong i ▁has ▁more ▁apples ▁than ▁N am j oon .
▁From ▁point ▁3, ▁we ▁know ▁N am j oon ▁has ▁fewer ▁apples ▁than ▁Y o ong i .
▁The ▁rate ▁of ▁the ▁current ▁is ▁given ▁as ▁8 .5 ▁kmph .
▁Each ▁package ▁of ▁inv it ations ▁has ▁3 ▁inv it ations ▁in ▁it .
▁First , ▁let ' s ▁calculate ▁the ▁score ▁for ▁the ▁H aw ks :
▁So , ▁we ▁will ▁consider ▁the ▁powers ▁of ▁2 ▁from ▁2 ^ 1 ▁to ▁2 ^ 5.
▁Vin ny ' s ▁initial ▁weight ▁was ▁3 24 .5 ▁p ounds .
▁The ▁list ed ▁price ▁is ▁$ 2 C , ▁so ▁the ▁cash ▁disc ount ▁is :
▁A th le te ▁2 ▁ran ▁10 ▁seconds ▁more ▁than ▁ath le te ▁1, ▁so ▁ath le te ▁2 ▁ran ▁for ▁55 ▁+ ▁10 ▁= ▁65 ▁seconds .
▁As ▁this ▁is ▁the ▁sum ▁of ▁the ▁ex ter ior ▁angles , ▁we ▁have
▁P ( A t ▁least ▁one ▁contract ) ▁= ▁4 /5
▁The ▁last ▁group ▁to ▁the ▁left ▁may ▁have ▁only ▁one ▁figure .
▁For ▁a ▁poly gon al ▁figure ▁in ▁one ▁of ▁the ▁plan es , ▁swe ep ▁it ▁through ▁space ▁in ▁a ▁direction ▁per pend ic ular ▁to ▁the ▁starting ▁plane ▁until ▁we ▁reach ▁the ▁end ing ▁plane .
▁0 .1 5 ▁* ▁( 30 ▁+ ▁D ) ▁= ▁number ▁of ▁green ▁d uc ks ▁from ▁both ▁po nds .
▁So , ▁T ony ▁can ▁lif t ▁90 ▁p ounds ▁in ▁the ▁cur l ▁exerc ise .
▁With out ▁additional ▁information ▁or ▁const ra int s , ▁we ▁can ' t ▁defin itive ly ▁determine ▁the ▁pattern .
▁This ▁means ▁that ▁in ▁5 ▁hours , ▁the ▁c row ▁e ats ▁N /4 ▁nut s .
▁Total ▁dem er its ▁so ▁far ▁= ▁12 ▁( f irst ▁week ) ▁+ ▁15 ▁( sec ond ▁week ) ▁+ ▁12 ▁( th ird ▁week ) ▁+ ▁5 ▁( f our th ▁week )