3 a 15a 6 b 21a 5 c 30a 6 d 12a 9. e 125a 8 f 36a 12 g 90a 13 h 56a a 6a b 5 c 3a 4 d 6a 4. e 10a 4 f 8a 2 g 5a 4 h 12a 2
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1 Answrs Cptr Workin wit surs Eris A Surs m ( + m Cptr Simpliin prssions usin t lws o inis Eris A Inis 9 i j 6 k 8 l 6 6 i j k l t i 9 ii i ii i 8 ii t m n 6 6 i j Eris B Sintii nottion i 8 j.8 k l.
2 , i A inst o multipli ii i Inor t minus sin in ront o ii i Got sins inorrt ii + 8 i + soul 6. ii km mils.6 6 rs. =. min. 9 tims.996 Cptr Workin wit lri prssions involvin pnsion o rkts Eris A Epnin rkts m k t + t + 6t + t 8m 9m k k 9 p + p 8w + w + i j r r + k + l m 6p + p + n 6t + 9t + o p + p m mn + n i + + j k
3 i Cptr Ftorisin n lri prssion Eris A Ftorisin ( ( t(m p t(t + ( ( + ( + + ( + + i t(6t + + j mt(t + m k ( + l pt(t + + p ( + ( + 6 (t + (t + (m + (m + (k + (k + 8 (k (k ( ( (m (m (p (p i ( + ( j (t + 8(t k ( + ( l (r + (r m (n 9(n + n (m (m + o ( 9( + 8 p (t (t + ( + ( + ( + ( ( + ( + ( + ( + ( + ( + ( ( + ( ( ( + ( i (8 + ( j (6 ( k ( ( + l (6 ( + ( + 9( 9 (t 6(t + 6 ( ( + (9 t(9 + t (k (k + (8 (8 + ( ( + ( ( + i ( ( + j ( ( + k (t w(t + w l (6 (6 + All t trms in t qurti v ommon tor o. ( + ( is t most omplt toristion s it s t ist ommon tor tkn out. 6 ( ( + ( ( + ( ( ( + ( ( + ( ( ( ( + ( ( Cptr Compltin t squr in qurti prssion wit unitr oiint Eris A Compltin t squr ( ( 6 ( + 9 ( 6 6 ( ( ( ( or + ( =, = =, = =, = 9 =, = = 6, = 6 =, = 6 =, = =, = ( + + ( + + ( + 6 ( 6 ( + ( 9 ( 8 + ( + i ( + 6 j ( + 8 k ( + 6 l ( 6 9 m ( + 6 n ( 9
4 o ( p ( q ( r ( s ( + + t ( =, = =, = =, = =, = =, = = 9, = =, = =, = i =, = j =, = 8 k =, = l =, = 9 m =, = o =, = q =, = n =, = p =, = r =, = 9 s =, = t =, = = +, = =, = = +, = = +, = = + 9, = 9 = +, = = +, = = +, = i = 6 + 6, = =., =. =., = 8.8 =., =. = 6., =.6 =., = 6. =., =. Cptr 6 Ruin n lri rtion to its simplst orm Eris 6A Simplst orm p 9q q t k m + + ( ( + j + ( + 6 l ( + ( + + k ( + = = 6 + ( ( + ( ( + 6 ( ( + ( ( + i + + ( + Cptr Applin on o t our oprtions to lri rtions Eris A prtions on lri rtions 6 8 ( + ( ( ( ( + t ( ( + i ( ( ( ( p ( = ( p p( p 8 p p + 9 ( ( ( ( + ( ( ( + + = +
5 ( + ( ( i = ( ( ( ( + ( ( (. m 6.8 m. mm 68 m 9. m 9 mm 88. m. m 6. m. mm 8 68 m 9 6. m m Cptr 8 Dtrminin t rint o strit lin, ivn two points Eris 8A Grint o strit lin Uns prt is 96.6 m. Cptr Clultin t volum o stnr soli Eris A Volum o soli 6. m 9.86 m.9 m 9. m 8 6 i j 9 k l 6 i 8 9 = 6 Ys, s t rint o t rmp is, wi is lss tn (or. <.66. Cptr 9 Clultin t lnt o n r or t r o stor o irl Eris 9A Ar lnt n stor r.9 m 6.8 m. mm. m. m.6 mm 6.8 m 8 mm 8. m. m 88 mm 6. mm 8.89 m.9 m.88 m 9 m 6. m 6 m 8 m. m m 6 9. m m m m m 8 m 9 86 m. m m.6 m Volum o on is 9 m, volum o prmi is 96 m, so on s ir volum n will ol mor i rm (9 > m 6.8 m
6 Cptr Rounin to ivn numr o siniint iurs Eris A Rounin to siniint iurs i j i j i 6 j 6 k 8 l 9. m 6 n o 8 p q.8 r.99 s.8 t i.69. k j l 9 Hll 8 99; Hook 6 6; Hunlton 6 A; t prkin sp is twn. n.8 mtrs lon n t r is twn. n. mtrs lon, so t sp is i nou.. m. m m 9 m Cptr Dtrminin t qution o strit lin, ivn t rint Eris A Eqution o strit lin =. + = =. =. =. rint =, -intrpt = rint =, -intrpt = rint =, -intrpt = rint =, -intrpt = rint =, -intrpt = rint =, -intrpt = rint =, -intrpt = rint =, -intrpt = i rint =, -intrpt = = + = = + = + = = = + = + = + 6 = + =.
7 = + = + = = 8 = = + = i = + ii m = iii (, i = ii m = iii (, i = ii m = iii (, i = ii m = iii (, i = + ii m = iii (, i = 6 ii m = 6 iii (, i = + ii m = iii (, i = + ii m = iii (, i i = 8 ii m = iii (, 8 j i = + 6 ii m = iii (, 6 k i = ii m = iii (, l i = + 6 ii m = iii (, 6 Eris B Funtions i 9 ii iii iv 6 i ii iii 8 iv i ii iii iv 6 v k = n k = i ii iii 8 iv 8 v vi. = n = i ii iii = n = Cptr Workin wit linr qutions n inqutions Eris A Solvin linr qutions = = = t = p = k = m = s = i w = = = t = = = 6 = t = t = t = 9 = p = = 8 = = = 9 6 Multiplin out t rkts ivs + = + 6, ivin = 6, wi is impossil. Multiplin out t rkts ivs + = +. Bot sis o t qution r t sm so oul n numr. Amr n Cllum ot tout o t numr. 8 rim, 8 sin ition, romn 9 Primtr is m ( =. Put n pir o sis qul,.. + = n solv. Solution =. Put into prssion or t sis: ll sis qul. So t nswr is s, t trinl is quiltrl wn =. = 8 m = 8 = 6 = 6 t = = = t = i w = 8 j = = 8 t = m = 6 p = = t = = = = = = = 9 = = = =
8 Eris B Solvin linr inqutions 9 6 < > > 8 < < >. > < < < > 6 t < < 6 > w < < 8 > > > 8 > < Cptr Workin wit simultnous qutions Eris A Solvin simultnous qutions rpill n lrill (, (, (, (, (6, (, 8 (, (, i (, j (, k (, l Lins r prlll so tr is no solution. =, = =, = =, = =, = =, = =, = =, = > =, = =, = =, = =, = =, = = 6, = =, = 6
9 6 =, = =, = = 6, = =, = =, = 8 CD., ook. 9 + = 8., + =. =., =. in ks n in pnuts.8. M minus ws oul m son s minus. = 6 n = + n =, + n = ivs =, n = 6. Totl ost or Mrus is., so will t.6 n. + p =, + p =, ivs = k, p = k T s wi k, so Crol nnot rr t s sl on r trilr. Cptr Cnin t sujt o ormul Eris A Cnin t sujt o ormul = k = p m = m 9 k 9 = or = k = A = = ( + = P R L L + Hj = + j or = H H t = = t or = J w = ( M k t = N p + = = M n = p + q = p q A st ( + = r = + r + t r = = s + A = = + = = + = = S B j 6 = = = K = Gm ( H k = = ( t C + p t L k = = jr p + Cptr 6 Ronis n trmin t qution o qurti untion rom its rp Eris 6A Equtions n rps o qurti untions k = k = k = k = k = k = k = k = k = k = k = k =. =, = =, = =, = =, = =, = =, = i =, = ii ( = ( + i =, = ii ( = ( + + i =, = ii ( = ( + i =, = ii ( = ( + i =, = ii ( = ( i =, = ii ( = ( + 6 =, = (
10 Cptr Sktin qurti untion Eris A Sktin qurti untion (, (, 9 6 (, (, 6 (, (, 9 (, 6 (, 6 (, 6 6 (, 6 6 (, (, (, 6 (, (, (, (, (, (, 9 (, 9 9
11 8 9 (, (, (, (, (, 6 (, (, (, (, (, (, 9 (, 9 (, 6 (, (, (, (, (, Cptr 8 Intiin turs o qurti untion Eris 8A Fturs o qurti untion i = ii (, iii minimum i = ii (, iii minimum (, i = ii (, iii mimum (, i = ii (, 8 iii mimum i = 8 ii (8, iii minimum i = ii (, iii mimum i = ii (, 9 iii minimum
12 i = 6 ii ( 6, iii mimum i i = ii (, 8 iii minimum i = ii (, 8 iii minimum i = ii (, 8 iii minimum i = 6 ii (6, iii mimum i = ii (, iii mimum i = 8 ii (8, 6 iii minimum i = ii (, iii mimum i = ii (, iii minimum i = ii (, iii mimum i i = ii (, iii minimum i = ( + + ii = iii (, i = ( + + ii = iii (, i = ( ii = iii (, i = ( 8 ii = iii (, 8 i = ( ii = iii (, i = ( + 6 ii = 6 iii ( 6, i = ( 6 ii = iii (, 6 i = ( + ii = iii (, ( i i = + iv = ( + + ii = ( + i = ( + iii = ( + v = ( + + vi = ( ii = iii Eris 8B Usin qurti untions to solv prolms 6 m sons sons (, No, s mimum it (t t = = s = m, wi is lss tn m. ( m ( = = A = m = P = Cptr 9 Workin wit qurti qutions Eris 9A Solvin qurti qutions torisin =, = t =, t = =, = =, = =, = t =, t = =, = =, = i = 6, = j =, = k =, = l =, = 6 =, = =, = 6 =, = =, = =, = =, = ( + ( + =, =, = ( + ( + 6 =, =, = 6 ( 8( + =, = 8, = ( ( + =, =, = ( + ( =, =, = ( + ( =, =, = (t 6(t + =, t = 6, t = (t 6(t + =, t = 6, t = i ( + ( =, =, = j ( ( =, = k (m (m =, m = l (t 8(t =, t = 8, t = m (t + (t + =, t =, t = n (k 6(k + =, k = 6, k = o ( ( 6 =, =, m = 6 =, = =, = 8 =, = =, = =, = = 6, = 6
13 ( + ( + =, =, = ( + ( + =, =, = ( + ( =, =, = ( + ( + =, =, = ( + ( + =, =, = ( ( =, =, = isriminnt = 8; two rl, istint roots isriminnt = ; two rl, qul roots isriminnt = 8; two rl, istint roots isriminnt = ; two rl, istint roots isriminnt = ; two rl, istint roots isriminnt = ; no rl roots i isriminnt = ; no rl roots 6 =, = =, = = 6, = =, = =, = =, = Eris 9B Solvin qurti qutions usin t qurti ormul =., =. =.9, =. =.9, =.9 =., =. =., =. =.9, =. =., =.8 =.6, =.6 =.69, =. =., =.8 =., =. =.9, =. =., =.6 =., =. =.8, =.8 =.8, =. i =.8, =.8 j isriminnt = ; two rl, qul roots k isriminnt = 6; no rl roots l isriminnt = ; two rl, istint roots 8 = ± Eri ts = 8 n Jun ts ( =. Bot in tt tr is onl on solution: =. T -is is tnnt to t urv. Eris 9C Usin qurti qutions to solv prolms Ell s rotr is ( + rs ol. ( + = + = + = =, so Ell is rs ol. ( + = ( + ( + wn pn, n lik trms ollt, ivs t rquir qution. = Ar = ( + ( = ( + ( = 6 m ( + m ( + = 6 6 = = 6 m, so lwn lnt = 6 m isriminnt = ; no rl roots isriminnt = 8; no rl roots
14 A = ( + m or A = ( + + m ( + ( = 8 Ars o squr n rtnl r t sm, so 8 = + +, ivin =. = 6, so lnt o squr = 8 m, lnt o rtnl = 6 m, rt o rtnl = m. Cptr Applin Ptors torm Eris A Applin Ptors torm Ys, s 9 + = 68 =. No, s + = 6, 6 = 66 n Ys, s + 8 = =. No, Jnn is not orrt s + = 9, 6 = 6 n 9 6. No, s 9 + =, = 6 n m. m DG =. m HA =. m DB =. m 8 6 m 9 m. m AG =. m AC = 9.9 m EX =.9 m EM =. m. m 9.9 m 6. m. m Ys, s + + =, n > (9. Cptr Applin t proprtis o sps to trmin n nl involvin t lst two stps Eris A Usin nl proprtis o irls Eris B Usin Ptors torm.8 m.9 m 9. m 6 m H = r. m 6. m. m Eris C Anls in polons 6 6 No, totl is. Ys, totl is 6. Ys, totl is 6. No, totl is. No, totl is. Ys, totl is 6. Pnton ivi into trinls, sowin 8 = 8 6 trinls 8 6 trinls 8 8 trinls 68 6
15 8 = 6, = = 9, = 9 = 8, = =, = 6 =, = m. m n. m Cptr Workin wit t rps o trionomtri untions Eris A Grps o trionomtri untions Anl AED = 8 (intrior nl o rulr pnton, so nl ADE = 6 (nls in n isosls trinl. Cptr Usin similrit Eris A Similr sps m.88 m m 9.6 m.9 m = 6.9 m, =. m = m, =. m m 6 No, Suzi is not orrt. T orrsponin sis r not in t sm rtio; or t trinls to similr, CD woul v to. m. = os = sin =.os = sin BC = m, CD = m 8 m m 6 m Eris B Ars n volums o similr sps m m m ( = 8 litrs 6. m T lr tin ols ml. H n ill smll tins. 6 6 m m %.% 8 iv 8 m 9 smll.9 m; lr. m mium 6 m ; smll. m = os = sin = sin = os
16 = os = sin = 6sin = os Eris B Trnsltions o rps o trionomtri untions 8 6 =, = (or = (or, = =, = (or = (or, = Cptr Workin wit trionomtri rltionsips in rs Eris A Solvin trionomtri qutions = 6.9,. =., = 66., 6. = 8.,. =.,.8 =.6, 6. = 9., 9. = 6.6, 9. =.,. =., =.,. = 9.,. = 8.,. =.8, 8. =., 9. =., 8. = os + = sin = os + = sin + i mplitu = ii prio = 6 i mplitu = ii prio = 8 i mplitu = ii prio = 8 i mplitu = ii prio = = 6.,.6 = 8.,.8 = 6., 6. =, =., 86.6 = 8.6,. = 6., 6. = 9., 8.
17 =.,. =.,. = 6.6, 96.6 =.6,. =.8,. =., 8. = 6.9, 6.9 =.,. = 9., 6. (.8, n (8., 6 m. s n 8. s m.6 m. p.m. n. p.m. Eris B Trionomtri intitis 6sin os tn LHS: + os = + ( sin = + sin = sin sin + sin os LHS: os ( = sin sin + os os = sin os = tn tn LHS: = tn sin sin = sin sin os = sin = os sin os LHS: sin + ( sin = sin + sin = sin Cptr Clultin t r o trinl usin trionomtr Eris A Clultin t r o trinl usin trionomtr. m.98 m 9. m m 8 m 89 m. m 9.m. m 9. m 6 m 8 m. m 6 m m 8 9 Cptr 6 Usin t sin n osin ruls to in si or nl Eris 6A Usin t sin rul. m. m 9(. (. 6 m 9 m 66(. (. (. 88 m.9 m m. m 8 9. m Ar o quriltrl = r o trinl ABD (6.8 m + r o trinl DCB (. m = 8. m Eris 6B Usin t osin rul 9. m.9 m Cos B = = =.86 m...6 m. m.6 km m 6. m Ys, s s nou pint, us. m = m, n m > m.
18 Eris 6C Coosin t orrt ormul 9. m.. 8. m m i 6. m m m ABC = 8. n is t lrst nl. Cptr Usin rins wit trionomtr Eris A Usin rins wit trionomtr. km.9 km 8 9 Eris 8B Usin vtors to solv prolms i ii + iii Bot r multipls o. 6. km 6.6 km Cptr 8 Workin wit D vtors Eris 8A D vtors p q p 8q q + p i ( ii + iii + T li on strit lin. 6 i ii T r prlll. i p + q ii p + q AB = CD = 6 p + q T li on strit lin. p = FE = q = r = Cptr 9 Workin wit D oorints Eris 9A Workin wit D oorints A (,, B (8,, C (8,, D (,, E (,, F (8,, G (8,, H (,,
19 J (,, K (,, L (,, M (,, N (,, 6 P (,, 6 Q (,, 6 R (,, 6 A (,, B (,, C (,, D (,, E (,, =, = A (,, B (6,, C (6, 6, D (, 6, E (,, 8 E (,, F (8,, G (8, 6, H (, 6, T (,, 9 K (,, L (,, M (,, N (,, P (,, Q (,, A (,, B (,, C (,, D (,, E (,, F (,, G (,, H (,, J (,, K (,, L (,, M (,, N (,, P (,, A (,, B (,, C (,, D (,, E (,, 6 F (,, 6 G (,, 6 H (,, 6 J (,, 9 K (,, 9 C (,, D (,, E (,, i (.,, ii (.,, iii (,,.9 units A (,, B (8,, C (8, 8, D (, 8, E (,, 8 F (8,, 8 G (8, 8, 8 H (, 8, 8 J (,, 6. units Cptr Usin vtor omponnts Eris A Usin vtor omponnts =, =, z = =, = Eris B Clultin t mnitu o vtors Cptr Workin wit prnts Eris A Prnt inrs n rs 9.. m 6. m. m 9. m Vronik., Amli.9, Srltt 8.6. Srltt s pon is wort t most rs i ii iii iv 6.6
20 Atr rs, t smor is 9. m tll n t onir is 9. m tll. Atr rs, t smor is. m tll n t onir is m tll. rs wks Eris B Rvrs us o prnts ours minuts 6 8 l % 8 % (still twi s mn Cptr Workin wit rtions Eris A Ain n sutrtin rtions i 96 9 j 69 k 6 99 l mils mils 9 To mk mtr pip, us two mtr pips n on mtr pip. Eris B Multiplin n iviin rtions 9 i 6 j k 6 l 9 6 i 8 j k 8 l i 8 9 j k l km 6 o (= is smllr tn o (=. 8 Ys, s 66 litrs wr out. 9 stions 8 ops tins Eris C Comintions o oprtions wit rtions
21 6 or or or 9 Cptr Comprin t usin sttistis Eris A Mn n stnr vition mn =, s =. mn = 9, s =.6 mn =, s = 8. mn =, s =. mn =, s = 8 mn =, s = mn =, s = 8 mn =, s = mn = k, s = 6.66 k n vr, plrs in Bton Am s tm r litr, n tir mss vris lss. mn =, s =.96 n vr, Jz s tm prorm ttr, ut R s tm ws mor onsistnt. mn = 6, s =. mn = 6, s =. n vr, tir sors wr t sm, ut Conni s sors vri mor. 6 mn =, s =.6 n vr, Hnritt prou mor s in t irst wk, ut r rsults wr lso mor vri in t irst wk. mn =, s =.6 n vr, t pupils prormn improv in t Dmr tst n tir rsults wr mor onsistnt. 8 mn = 86., s =. T ition o. ookin inrs t mn. ut i not t t stnr vition, us t pri o oli inrs t sm mount. Eris B Min n smi-intrqurtil rn i min = 6. ii Q =, Q = 8, SIQR =, SIQR = i min = 8 ii Q =., Q =, IQR =., SIQR =. i min = ii Q =, Q =, IQR =, SIQR =. min = rs, IQR =. rs, SIQR =. rs min =. rs, IQR = rs, SIQR = rs n vr, t s o trs in t musi prtmnt r ir ut lss vri. min = 9, IQR =, SIQR = 6 n vr, t tors rn mor ut tir slris wr mor vri. min = min s, SIQR =. s n vr, t sons wr sortr on t irst lum, ut t lnts wr mor vri. min =.68, SIQR =. n vr, t quots or Nw York wr mor pnsiv n mor vri. 6 min =, SIQR = n vr, t ru tm omplt mor prss-ups pr minut n tir rsults wr lss vri. min = min s, SIQR = 6 min s n vr, t stunts took lonr to omplt t istor tsks, ut tir tims wr mor onsistnt.
22 8 mn = 66%, s =.8% min = 69%, SIQR = % Usin t min n SIQR is ttr us r rsult or Art is n outlir., D Cptr Formin linr mol rom ivn st o t Eris A Drwin n usin st-ittin lin rom ivn t, Distn (km 6 Tim (min 8 T 8 Usin (, n (,, D = T Musi m 6 ppro. 6 km ppro. 8. min D = 9T mils 6 8 Mts m ppro. ours 9 min P = M + 8 Usin (, n (, 6: = + (to nrst wol numr 9 H = F + m m 6 L = -.R ours ours N = T + 6 pirs o lovs
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