Getal en Ruimte 1VWO deel 2

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1 Gtal n Rumt 1VWO l 2 MA2N Hoostuk 8 Utwrknn m / 2

2 Gtal n rumt 1VWO l 2 - Hst Hrln 1a 6a + 2a = 8a 7p p = 6p 3p + 3q = kan nt 4a 7 = 28a 2a 5a = 10a² j 2 3a = 6a 3a a = 8a + 2 2a a = a + 2 8a + 3a + a = 12a 2a a 4 = 2a 5 3a = 15a 6 2a = 12a 5 3a + 6 2a = 27a 3a = = = 31 5a 2 2a = 8a 5a 3 2a = 15a + 2a 8a 3 5 3a = 9a 4a 3 2x x = 14x 5 3y 8 2y = y 5x 2y 15x y = 5xy 3x 2z 5x 2y = 6xz 10xy 4 2x 8 6 = 8x x 8x 3 = 16x 5a 2 3x x = 16x 2 + 3x x = 5x x x = 8x x + 5x 2 x = 3x 3 + 2x x = 2 3 2x 5 2 x = 16x 6a 3 4a + 5 2a = 22a 3a 4 + 2a 3 = 18a 3a a + 3 = a + 7 3a 3 + 2a 3 = 3a 3a 3 + 2a 3 = a 6 3a 3 + 2a 3 = 3a 7a 16a = 4a + 12a 1 2 8xy = 2 xy 5 xy a = 17a a 1 16a = a 64a 4 18xy = 2x 3y + 3x 4y 8xy = 2x 3y + x 2y 8 15a 3a = 12a 2a 3 + 3a 2 = 12a 3 4a = 12a 5a 4 2a 4 = 12a 1 1 8a = 12a 2 9a 2 + 6a 5 = 12a a 12 = 12a 10a + 2a = 12a 4a 2 + a 2 + 2a = 12a a + 9 a = 12a 3 3 9a 5x 3y 3x + 2y = 15xy 3x + 2y 5 3y 3x 2y = 15y + 6xy 5x + 3y 3x 2y = 2x + y 6a 2 2 a + 5a 2 = 20a 3a 2 2 a + 5a 2 = 3a 2 8a 3a 2 2 a + 5a 2 = 7a 6

3 10a D pottn n oosjs op plank 4 vattn 2a + 3 kraln. D kast vat n totaal 12a + 5 kraln. Ht totaal aantal kraln s 5a + 3 3a + 2 = 2a + 5 kraln. 11a 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, 898, Ht jzonr aan rj van Fona s at rj nt j t jr 1. (Soms laat mn rj van Fona ook wl nnn mt 0,1,1,2.) a,, a +, a + 2, 2a + 3, 3a + 5, 5a + 8, 8a + 13, 13a + 21, 21a + 34, a +, 2a, 3a +, 5a +, 8a + 2, 13a + 3, 21a + 5, 34a + 8, a, a +, 2a, 3a +, 5a +, 8a + 2, 13a + 3, 21a + 5, a 2, 2a, a, 3a, 4a, 7a, 11a 2, 18a 3,

4 Gtal n rumt 1VWO l 2 - Hst Haakjs wwrkn 12 D omtrk van rtok kun j srjvn als a + + a + o 2a + 2 o 2(a + ). 13a Totaal aantal plantn = 7 5 Aantal plantn onr nt = 4 5 Aantal plantn nt onr nt = 3 5 Ht totaal aantal plantn s ljk aan t aantal plantn onr t nt plus t aantal plantn nt onr t nt. Dus 7 5 = a 5(a + ) = 5a + 5 8(2a + ) = 16a + 8 5a(2 + ) = 10a + 5a 3p(x + 3) = 3px + 9p 1½(4x + 2y) = 6x + 3y 8p(2r + s) = 16pr + 8ps 15a 1 6 a(3 + ) = 18a+ 3a ( a + 6) = a p(4q + 8 s) = 2pq + 4ps a( + 14 ) = 7a + 12 a 5 a(2 + 1 ) = 10a + 7 a 2 1 pq(3r + 1 s) = 2 pqr + pqs a 4(a + 3) + 2a = 4a a = 6a (x + 2y) + 3y = 5x + 10y + 3y = 5x + 13y 5(a + 2) + 3(3a + ) = 5a a + 3 = 14a +13 4(2x + 5y) + 5(x + 5y) = 8x + 20y + 5x + 25y = 13x + 45y 17a y = 8(x + 6) = 8x + 48 T = 4(2a + 3) = 8a + 12 y = 5(x + 1) + 3(2x + 6) = 5x x + 18 y = 11x +23 A = 5(2 + 3) = a 4(x + 2y) = 4x 8y 4(x 2y) = 4x 8y 4(x 3) = 4x (2x + 8) = 8x 32 (2x 3) = 2x + 3 (2x + 3) = 2x 3 19a 3a(2 + ) = 6a 3a ( 4 q 3r) = q+ 12r 5a(3 ) = 15a 5a ( 3p q) = 3p + q 2p(3q 1) = 6pq + 2p ( 1 ) = a 3(a + 2) 6a = 3a + 6 5(a 2) + 6a = a (a 2) + 3(2a ) = 11a 13 8(a ) 5(a 3) = 3a a (5 + 2a) = 5 3a( 1) 3a = 3a + 3a 3a = 3a

5 21a 7(x + y) 2(3x y) + 5x = 7x + 7y 6x + 2y + 5x = 6x +9y 3p p(q + 3) + q(2 p) = 3p pq 3p + 2q pq = 2q 2pq 3(a 6) 2a a( 4) = 3a a a + 4a = a a 8y 2x(y +7) y(x 8) = 8y 2xy 14x xy + 8y = 16y 3xy 14x 22a 3x 2y + x(y 5) = 8x 2y + xy 5p(q + 3) 2p 3q = 11pq + 15p 3a 7 + 2a(3 5) + (1 a) = 21a + 6a 10a + a = 10a + 6a + 2x(5y 2) 5x 2y = 4x 2p(q 1) (1 2p) = 2pq 2p 1 + 2p = 2pq 1 3p 2q q(6p + 1) = 6pq 6pq q = q 23 Blauw: Opprvlakt = x(y + 3) = xy + 3x Omtrk = 2(y + 3) + 2 x = 2y + 2x + 6 Gl: Opprvlakt = 7(p + 5) = 7p + 35 Omtrk = 2(p + 5) = 2p + 24 Gron: Opprvlakt = 2p(q ) = 2pq 2p Omtrk = 2(q ) + 2 2p = 2q 2 + 4p 24a Er zttn a 16 kraln n pot. Franka rjt (p 2) 3 = 3(p 2) = 3p 6 klomtr. Wllk rjt 12 a klomtr. Japp krjt 4(p + 5) = 4p Flors was 5p 5 jaar. Tomas zjn var s 4(x 3) = 4x 12 jaar. 25a J krjt (x +2) 3 = 3(x + 2) = 3x + 6 J krjt 4(a + 5) 4a = 20 26a Lnt = 4(a + 3) + 5(2a + 4) = 4a a + 20 = 14a + 32 Lnt = 8(a + 3) + 3(2a + 4) + 7(3a + 8) = 8a a a + 56 = 35a + 92 Lnt = 32a (a + 3) = 32a a 9 = 29a +87 Lnt = 40a (2a + 4) 4(3a + 8) = 40a a 8 12a 32 = 24a a - Ht kozn tal kun j truvnn 24 van utkomst a t aln. (x + 6) 4 3 x = 4(x + 6) 3x = x + 24 D utkomst s us 24 mr als t kozn tal. Nm n tal n atn, vrmnvul t mt 2, tl r 5 j op n vrmnvul z utkomst mt 4. Nu van t rsultaat 8 kr t oorspronkljk tal ra aln. D utkomst s altj 20. Brknn: (2x + 5) 4 8 x = 4(2x + 5) 8x = 8x x = 20

6 28 I Is just, ljkt r op ornl rl. II Is just,.p.v. vrmnvuln nu ln. III Is nt just, 12 : (2 + 4) = 2 n 12 : : 4 = 9 IV Is nt just, 10 (3 + 4) = 3 n = 13 V Is nt just, 20 (15 : 3) = 100 n (20 15) : (20 3) = 5 VI Is just, s n utrn van rl a( + ) = a + a VII Is just, ook wl kn oor rl: ln oor n ruk s vrmnvuln mt t omkr

7 Gtal n rumt 1VWO l 2 - Hst Matn 29a Woutr t an = 64 snpprs. Ik sat at j an ook 6 kr kunt surn, oout 7 kr want anrs krj j tvl snpprs om n één kr oor t surn. 30a 5³ = = 125 5³ sprk j ut als vj tot mat 3. In 5³ s 5 t rontal n 3 xponnt. 31a 2 5 = = = = = = = = 0 32a 5 4 = = = = a 8 = = = = = = = = = 13 2 j k l 1 = = = a Exponnt utkomst , 2 23 n 2 55 nn op n , 2 8, 2 12, 2 16, 2 20, t nt op n 4. ( 2 94 = ) 9 23 = , nt us op n a = = = = 117 (5 2) = = 83 (3 4) 2 8 = = = = 7 ( ) 2 = 11 2 = 121 j k l = = : 3 2 = 36 : 9 = 4 5 (3 2) 3 = = 5 24 = 19 37a 96 = = = = = = = = = = = = a = = 16 Ht rst tal = 16 s vr mat van 2.

8 39a ( 2) 4 = = 16 ( 3) 3 = = ( 2) 4 = = 11 ( 1) = ( 2) 3 = 20 40a ( 4) 6 s pos. ( 11) 19 s n. ( 5) 7 s n. ( 1,5) 12 s pos. ( 0,3) 9 s n. ( 7) 10 s pos. ( 1) 10 s pos. ( 12) 15 s n. Rl: Nat rontal n vn xponnt an s utkomst post; Nat rontal n onvn xponnt an s utkomst nat. D rl lt nt voor 7 4 want t s nt vr mat van 7 41a 5 ( 2) 4 = 5 16 = ( 2) 5 = = 7 (5 2) 4 = ( 10) 4 = (2 3)6 = 5 1 = ( 3) 4 = = ( 1) 5 = = 2 32 : ( 2) 5 = 32 : 32 = ( 3) 2 = = = ( 2) 3 = = = 47

9 Gtal n rumt 1VWO l 2 - Hst 8 42a 2,3 2,3 2,3 2,3 = 27,9841 2,3 4 = 27,9841 D utkomstn zjn ljk! 8.4 Matn n rknman 43a 5,6 3 = 175,62 0,95 4 5,2 = 4,24 ( 2,6) 4 1,7 3 = 40,78 0,58 2,5 3 ( 1,4) 3 = 11,81 (3,7 0,27) 5 = 1,00 (4,7 2,8) 3 : ( 1,8) 4 = 0,65 44a 3 4 (1 7) 4,16 3 2,1 8 4 = = 0,61 1, (2 3) + ( 1 6) = 14, : ( ) 211,64 4 = 7 5 ( 2,3) 100 = 10, = 1, ,3 5 2,8 45a In 2003 wr r ,7 = ton aanvor n n 2004 was at ,7 = ton 2008 s 6 jaar na 2002, j mot an us ,7 0,7 0,7 0,7 0,7 0,7 utrknn. Dt s tzl als ,7 6 = 5882,45 ton. Dt s ,45 = 44117,55 ton mnr als n 2002!!!! 46a Joan Ptrs a 16 ouovrrootours ( 2 4 = 16 ). Ht aantal voorours van 10 nrats tru s 2 10 = s 510 jaar ln. Dat zjn 510 : nrats ln. Dus 2 20 = voorours = : nrats 2 32 = voorours 47a 10 5 = Er staan 5 nulln n t antwoor. Bj 10 9 krj j 9 nulln. En mljon t 6 nulln. 1 mljon = = mljar = = a = 4, = = 1, = 1, = 1, = 5, a En volwassn olant kan 6, ram wn. En uw uurt jna 3, sonn. In n mnsnlvn slaat t art onvr 2, kr. Ht aantal arn op n oo s onvr 1, D lauw kra lt n aar lvn 1, rn. 50a 5, = , = , =

10 = = = a 2 58 = 2, ,3 28 = 1, ,8 3,7 25 = 2, ,5 14 3,1 9 = a ( ) : ( ) = 1280 rvn passn op USB-stk. ( ) : ( ) = 123,0769. Er passn us 123 -roms op vast sj. 54a = 16 1 = = = 32 1 = 31 Voor rst 10 vakjs: = = 1023 raankorrls. Voor rst 20 vakjs: = = raankorrls. Voor t l saakor warn at = 1, raankorrls. (1, ) : = 1, waons van 10 mtr no. Er aan 100 waons n n klomtr, us 1, : 100 = 1, klomtr. 1, : = D trn zou us kr omtrk van aar worn!! Sssa vro 1, : = 1, ton raan Dat s 1, : 10 9 = 1025 mljar ton raan. Ht uurt us 1025 jaar voorat aan sn wns van Sssa s volaan. 55a 0, : = 5, komt n t vnstr van rknman = 10 komt r n t vnstr a 0,0037 = 3, , = 8, ,07183 = 7, ,00007 = ,12 = 1, = 8, a 0, = 3, , = 9, , = a 3, = 0, , = 0, , = 0, a 0, = 2, ,7 0,12 24 = 6, ,21 8 0,035 7 = 2, ,32 20 = 1, = 9, :8,5 = 3,

11 60a 5x 2x = 10x 2 4x x 3x = 12x 3 2a 5a 3a a = 30a 4 Gtal n rumt 1VWO l 2 - Hst 8 5p p p = 5p 3 4x x = 4x = Matn n lttrs 61a p 3 p 4 = p 7 p 5 p 3 = p 8 62a 5 8 = 13 x x 4 = x 5 x 3 x 5 x = x 9 x 5 x x x 2 = x 9 p 8 p 9 = p 17 p p p = p 3 63a 2a 5 3a 7 = 6a 12 8a 6 3a = 24a 7 4m 6 2m 7 = 8m 13 y 3 2y 9 = 2y 11 3p 4 2p 12 = 6p 16 q 6 3q = 3q a 13 54a 13 96a 20 4a 7 8a 6 18a 5 3a 8 8a 8 12a 12 2a 4 2a 3 4a 3 6a 3a 4 a 4 2a 5 4a 3 3a 9 a 3 2a a 2 4a 2 3a a 3 a a 3 2a 2 2a 1 a a 5x 3 2x 4 x = 10x 8 a 3 a 6 2 = a 7 5 x 2 y 3 x 5 y 2 = x 7 y 5 2a 3 5a 3 2 = 10a 4 8 3p 6 3p 6 = 9p 12 6x 2 3y 2y 5 2x = 72x 3 y 6 66a Luuk t natuurljk ljk want j mot j n vrmnvuln xponntn j lkaar optlln = = = a Inou = zj zj zj = a a a = a 3 Opprvlakt rnsvlak = a a = a 2 Opprvlakt kuus = 6 a 2 = 6a 2 68a 7a 3 + 5a 3 = 12a 3 3a 2 9a 2 = 6a 2 4a 4 a 4 = 3a = = = 15 2 a 2 + 2a 2 = 3a 2 x 2 + x 2 = 2x 2 2x 2 + 8x 2 = 10x 2 69a 8a 6a = 2a 13p 2 + p 2 = 14p 2 5a 3 + 5a 3 = 10a 3 6x 6 7x 6 = x 6 2x 2 + 8y 2 kan nt 8a 2 7a 2 = a 2 6a 2 10a kan nt 5x 2 5x 2 = 0 7p p 3 = 48p 3 70a 3x 5 + 2x 5 = 5x 5 3x 5 2x 5 = 6x 10 5x 3 2x 5 kan nt 5x 3 2x 5 = 10x 8 x 3 + 2x 3 = 3x 3 x 3 2x 3 = 2x 6

12 71a 12x 6 = 8x 6 + 4x 6 12x 6 = 6x 4 2x 2 12x 6 = x 12x 5 12x 6 = 15x 6 3x 6 12x 6 = 2x 4 6x 2 12x 6 = 3x 4x a 5 9a 3 = 3a a 8 = 87a 8 69a 8 = 6a 3a 7 = 5a a 8 = 10a 8 28a 8 = 18a a 7 = 8a a 8 = 19a 8 a 8 = 6a 6 3a 2 = 73a a 3 a 2 + 3a 5 = 4a 5 2a 4 a 2 + 5a 3 a 3 = 7a 6 7a 8 + 2a 2 3a 6 = 13a 8 8a 2 3a 5 2a 4 3a 3 = 30a 7 74a 3a 2 (a 4 + 2a) = 3a 6 + 6a 3 5a(a 3 2a) = 5a 4 10a 2 3a 2 (a 2) = 3a 3 + 6a 2 a 3 (a 5 a 4 ) = a 8 a 7 a 3 (2a 1) + a 3 = 2a 4 a 3 + a 3 = 2a 4 a 2 (a 3 2a) + a 4 (a 1) = a 5 2a 3 + a 5 a 4

13 75a Gtal n rumt 1VWO l 2 - Hst 8 2x 2x 2x = 8x 3 2x 2x 2x 2x = 16x 4 Ja (2x) 4 = 2x 2x 2x 2x 8.6 Hrln van Matn 76a (a) 7 = a 7 7 (pq) 12 = p 12 q 12 ( 4x) 2 = 16x 2 (a) 9 = a ( 3a) 3 = 27a 3 ( 2pq) 4 = 16p 4 q 4 (4xy) 2 = 16x 2 y 2 (10a) 3 = 1000a 3 ( a) 3 = a a (3a) 2 ( 2a) 3 = 9a 2 8a 3 = 72a 5 3 (3) 3 = 27 6 ( 4) 3 ( ) 5 = 64 8 ( 2a) 6 (5a) 2 = 64a 6 25a 2 2 = 1600a 8 2 (3xy) 4 xy 2 = 81x 4 y 4 xy 2 = 81x 5 y 6 ( 2) = = (4a) 2 = 16a a 2 2 ( 4a) 2 = 16a a 2 3 4a 2 9 (2a) 2 = 4a 2 4 8a 2a = 16a a 4a = 16a 2 5 2a 2a = 4a 2 11 ( 2a) 2 = 4a 2 6 (4a) 2 = 16a a 2 D orjs 1, 2, 7 komn op tzl nr. D orjs 3 n 5 n orjs 4, 6, 8 n 10 n vrr orjs 9, 11 n 12 komn tlkns op tzl nr. 79a Inou kuus I = 4a 4a 4a = 64a 3 Opprvlakt kuus I = 6 4a 4a = 96a 2 Inou kuus II = 8a 8a 8a = 512a 3 Opprvlakt kuus II = 6 8a 8a = 384a 2 512a 3 : 64a 3 = 8 D nou van kuus II s 8x zo root als nou van kuus I. 384a 2 : 96a 2 = 4 D opprvlakt van kuus II s 4 x zo root als opprvlakt van kuus I. 80a (5 2 ) 3 = 5 6 (3 2 ) 4 = 3 8 (10 3 ) 4 = (2 2 ) 2 = 2 4 (6 6 ) 2 = 6 12 (7 4 ) 4 = a (a 2 ) 6 = a 12 ( 5 ) 2 = 10 (a 2 ) 3 = a 3 6 ( 8 ) 2 = 16 (a 7 ) 4 = a 28 (p 6 ) 8 = p 48 a 3 (a 9 ) 2 = a 3 a 18 = a 21 x (x 12 ) 10 = x x 120 = x 121 (p 3 ) 4 (p 6 ) 3 = p 12 p 18 = p 30 82a (2a 2 ) 3 = 8a 6 ( 3a3)3 = 27a9 (a2)3 = a35 (x 3 y) 2 = x 6 y 2 ( 3a 2 ) 4 = 81a 8 4 (4a 2 2 ) 3 = 64a 6 6 (a 3 3 ) 4 = a ( xy 6 z 5 ) 4 = x 4 y 24 z 20 (pq 3 ) 2 (p 6 q 3 ) 3 = p 2 q 6 p 18 q 9 = p 20 q 15 83a x x x 3 x = 11 x x = 2 18x 2x 9x = 7 15x 8 3x 8 = 5

14 84a a a 3 4 a = 3 21a 3a 7a = a 12 a = 2 85a a 8 5 a 3 a = a8 a 3 = a 11 a 8 + a 3 a 2 = a 8 + a 5 (a 3 ) 8 = a 24 8a 3 3a 8 = 24a 11 (3a 8 ) 3 = 27a 24 86a (2a) 3 + 4a 3 = 8a 3 + 4a 3 = 12a 3 (2a 2 ) 3 + (3a 3 ) 2 = 8a 6 + 9a 6 = 17a 6 4a 3 2a 5 + (3a 4 ) 2 = 8a 8 + 8a 8 = 16a 8 2x 3 x + ( 3x 2 ) 2 = 2x 4 + 9x 4 = 7x 4 ( 5x 4 ) 3 + 5(2x 3 ) 4 = 125x x 12 = 205x 12 (5a 3 ) 2 3a 4 ( 2a 5 ) 2 = 75a 10 25a 10 =100a 10 87a 3a (4a) 2 a = 3a a 3 2 = 19a 3 2 (2xy) 3 2(xy) 3 = 8x 3 y 3 2x 3 y 3 = 6x 3 y (3 a) 9a 2 18 pq = = a = 6q 6 3 9a 9a 3pq 3(x 2 y) 3 (5xy) 2 x 4 y = 3x 6 y 3 25x 6 y 3 = 22x 6 y 3 a 2 (3a 4 5) + 5a 2 (2a 3 ) 2 = 3a 6 5a 2 + 5a 2 8a 6 = 5a 6 88a (5a 3 ) 4 3a 2 2a (a 9 ) 2 7(a 3 ) 2 (2a 4 ) 3 + ( 9a 9 ) a 18 = 625a 12 6a a 18 7a 6 8a a a 18 = 3750a a a a a 18 = 6928a 18 7(a 3 4 ) 6 18(a 9 ) 2 (2 6 ) 4 + (3a) 2 (2a 3 4 ) 2 10(a 5 7 ) 2 + 5(a 3 3 ) 2 13(a 2 3 ) 6 = 7a a a 2 2 4a a a a = 7a a a a = ( 2 a) + 4( a) ( a) ( 3 a) 2( a) ( a ) ( ) 64a + 4a a + 27a 2a = a a 4a 54a 6a = = = a 6a

15 Gtal n rumt 1VWO l 2 - Hst 8 Gmn Opavn