Matematika 3 Razred Gimnazije Banja Luka

8/9/2019 Matematika 3 Razred Gimnazije Banja Luka

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. : III

1

Маака IIIа ћ а ај

Рак Вкћ, .

2012-13. .. .


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. : III

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Садржај

I .................................................................................................................. 5

1. ......................................................................................................... 5 1.1. ................................................................................................ 5

1.2. .......................................................................................... 7 1.3. ............................................................................................ 8 1.4. ......................................................................................................... 9 1.5. .......................................................................................................... 10

2. ............................................................................................................ 12 2.1. .............................................................................................................. 13 2.2. ........................................................................................ 14 2.3. ...................................................................................................... 16

3. .......................................................................................................................... 17

3.1. .......................................................................................................... 17

3.2. ................................................................................................................ 20 4. () ....................................................................................................... 21

4.1. .............................................................................................. 22 4.2. ...................................................................................................... 22 4.3. ....................................................................................................... 23 4.4. .................................................................................. 23

5. ........................................................................................................................ 24 5.1. .............................................................................................................. 25 5.2. ............................................................................................................ 27 5.3. ................................................................................................................. 29

II ...................................................................................................................... 31

6. ...................................................................................................................... 32 6.1. .............................................................................................. 32 6.2. ............................................................................................ 34 6.3. ...................................................................................... 35 6.4. ......................................................................................................... 37

7. ................................................................................................................. 38 7.1. ........................................................................................ 39 7.2. ........................................................................................ 41

8. ............................................................................................... 45 8.1. ................................................................................................................. 46

8.2. ..................................................................... 47

8.3. ..................................................................... 48 9. ................................................................................................ 50

III ............................................................................................................. 57 10. ..................................................................................................................... 58

10.1. .................................................................................................. 59 10.2. ............................................................................................ 61 10.3. ............................................................................................ 63


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11. ........................................................................................................................ 65 11.1. ........................................................................................... 67 11.2. ............................................................................................... 72 11.3. .............................................................................................. 74

12. ..................................................................................................... 76

12.1. ............................................................................ 77

12.2. ........................................................................... 79 13. ...................................................................................................................... 82

13.1. ............................................................................................. 83 13.2. ............................................................................................. 85

IV ....................................................................................................... 91 14. .............................................................................................. 91

14.1. .................................................................... 91 14.2. ........................................................................ 95 14.3. ............................................................................................. 99

15. ...................................................................................................... 102

15.1. 2. ................................................................................... 102

15.2. ........................................................................... 107 16. ................................................................................................................ 114

16.1. .......................................................................................... 114 16.2. ........................................................................ 118 16.3. ....................................................................... 123

17. ............................................................................................................... 127 17.1. .......................................................................................... 127 17.2. ........................................................................................ 132

V ................................................................................................................ 137 18. ....................................................................................................... 137

18.1. ..................................................................................... 138 18.2. ............................................................................................ 140

18.3. ........................................................................................... 144 18.4. ................................................................................. 151 18.5. ............................................................................. 153

19. ........................................................................................... 157 19.1. ............................................................................................ 157 19.2. ........................................................................ 158 19.3. ....................................................................... 160

VI ........................................................................................... 165 20. ............................................................................................................. 166

20.1. .................................................................................... 166 20.2. ........................................................................................ 170

20.3. ............................................................................................ 175 21. .................................................................................................................. 179

21.1. .......................................................................................... 180 21.2. .............................................................................................. 183

22. .............................................................................................................. 190 22.1. ...................................................................................... 190


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4

22.2. .......................................................................................... 193 23. ............................................................................................................ 199

23.1. .................................................................................... 200 23.2. ....................................................................................... 207

VII ..................................................................................................................... 213

24. ..................................................................................... 213

24.1. ............................................................................................... 213 25. ................................................................................................... 224

25.1. .......................................................................................... 225 25.1. ......................................................................................... 226

26. ....................................................................................... 228 26.1. .................................................................................... 229 26.2. .............................................. 232

VIII ............................................................................................................. 236 27. ...................................................................................................... 236

27.1. .............................................................................................. 237

27.2. ............................................................................................... 240

27.3. ......................................................................................................... 245 27.4. ............................................................................................................. 249 27.5. ......................................................................................... 251

28. ........................................................................................... 257 28.1. .................................................................................. 258 28.2. ........................................................................................ 261 28.3. .................................................................................................... 265


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. : III

5

I

1. .

c

a

sin , c

b

cos , tg = b

a

, ctg = a

b

.

:

1cossin 22 , tg =

cos

sin, ctg =

sin

cos,

cossin , tg = ctg . ђ

21

1cos

t ,

21

sint

t

,

t = tg , 0-90 +.

1.1.

I . .

1.1.1.

1. : ) ; ) .

2. , :

30 45 60

sin2

1

2

2

2

3

cos2

3

2

2

2

1

tg3

3 1 3

, .


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6

sinAD - ,

cosOA - ,

CD = tg - ,

DE = ctg - ,

OC = sec cos

1 - ,

OE = csec sin

1 - .

, :

AB = versin = 1 – cos ,

BC = exsec = sec - 1.

a, b c ABC , . r R.

1.1.2. .

Rcba

2sinsinsin

: , ABC

2R = A1C. A1BC

, 1 BC

. R

a

2sin , . R

a2

sin

.

Rb 2sin

Rc 2sin

,

.

1.1.3. :

cos2222 bccba ,

cos2222 accab ,

cos2222 abbac .

: , h = CD

ABC c x y.

22222 xbyah , xcy 22222 )2( xbxcxca cxcba 2222 . cosbx ,

. , .


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7

1.2. Ј

, .

1( ) .

, .

OA = cos - , OF = sin - , BC = tg , ED = ctg .

(BG)

(BOG) . 2 6,28319,

360. 2 = 360. = 180, 902

,

603

, 45

4

, 30

6

, . .

OB, OG . ( BE)

0 2

, II 2

, III 2

3

, IV 2

3

2 . , II , , .

(x) f(-x) = f(x). (x) f(-x) = -f(x). , , .

1.2.1.

1. :

I II III IV

sin + + - -cos + - - +

tg + - + -

ctg + - + -

2. , .


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8

3. , .

:

DAC . ?

1.3.

1.3.1. , :

sincoscossin)sin( .

:

AD

EDGE

AD

GD)sin(

AD

EDBC

AD

CD

CD

ED

AD

AC

AC

BC sincoscossin .

1.3.2. , : ) sincoscossin)sin( ;

) sincoscossin)sin( .

)

2

1R , (

1.3.) a sin , b sin c sin .

; AEO: cc

2 / 1

2 / sin , .

, coscos abDBADc

( ) cossincossin . sinc ))(sin(

)sin( , . ) () ))(sin(

)sin(cos)cos(sin sincoscossin ,

, .

1.3.3. , : sinsincoscos)cos( ,

, .


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: , ))(2

sin()cos(

))2

sin((

cos)2

cos(cos)2

sin( sinsincoscos .

))2sin(()cos(

sin)2cos(cos)2sin( sinsincoscos .

1.3.4. :

)

tgtg-1

tgtg)(tg

; )

tgtg1

tgtg)(tg

.

1.4.

( -) .

, xx 2

21

, .

1.4.1.

1. :

) cossin22sin ; ) 22 sincos2cos ; )

2tg1

tg22tg

.

2. :

) sin3sin43sin 3

; ) cos3cos43cos 3

.

3. :

) 2

cos1

2sin

; )

2

cos1

2cos

;

1.4.2. 2

tg

t :

) 21

2sin

t

t

; )

2

2

1

1cos

t

t

; )

21

2 tg

t

t

.

: ,

2

, ta 2 , 21 tb ,

222 cba 21 tc . ,

.

1.4.3.


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10

1. : ) )sin()sin(cossin2 ; ) )sin()sin(sincos2 ;

) )cos()cos(coscos2 ; ) )cos()cos(sinsin2 .

: )sin()sin( )sincoscos(sin)sincoscos(sin cossin2 , .

2. :

) 2

cos2

sin2sinsin

; ) 2

sin2

cos2sinsin

;

) 2

cos2

cos2coscos

; ) 2

cos2

sin2coscos

.

: yx , yx

2

cos2

sin2 2

)()(cos2

)()(sin2 yxyxyxyx )cos(sin2 yx

yxcossin2 )sin()sin( yxyx sinsin , .

1.5. Ј

Ј ( -) . ,

. , 532 x ,

1x .

1.5.1. :

) 2

2)60sin( x ; )

2

1)

6cos(

x ;

) tg(x – 30) 3 ; ) ctg3

3)

4(

x .

: ) () ђ 707,02

2

() ( ). ()

ђ 45 135. + 60 = 45, + 60 = 135. () k , . , :

+ 60 = 45 + k 360 + 60 = 135 + k 360. 360151 k x 360752 nx (k,nZ). ) ,


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12

222222cossin

ba

cx

ba

bx

ba

a

.

22 ba

a

22 ba

b

,

1. cos22

baa , sin

22

bab ,

22cossinsincos

ba

cxx

, .

22)sin(

ba

cx

.

, 1122

ba

c.

, 1.

1.5.5. ABC 5

4sin , 135sin .

cos 2 tg

, .

: ))(cos(cos )cos( )sinsincos(cos

13

5

5

4

13

12

5

3

65

56

65

2036

.

3

4 tg

3

4

1

22

t

t ( 1.4.2.),

2

tg

t , 0232 2 tt

2

1t 2t . ,

2

1

2 tg

.

2.

a b P = ab. .

baP haP ahP2

1

. :

1. , 2. , 3. .


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13

.

2.1.

ABD CBE

, B. AB:AD = BC:CE, . c:ha = a:hc . a:b:c =hc:hb:ha , .

2.1.1. ABC:

222

cba hchbhaP

.

2.1.2. sin cha , sin2

acP

.

, ђ .

2.1.3. :

))()(( csbsassP , 2

cbas

.

, ( 1.4.) ac

bca

2cos

222

sin2

1acP 2cos1

2

1ac

222222)(4

4

1bcaca

)(2)(241 222222

bcaacbcaac 2222

)()(4

1

bcacab

))()()((4

1bcabcacabcab ))()(( csbsass .

2.1.4. .

2.1.5. R

abcP

4 , R .

( 1.3.) 2.1.2.

2.1.6. rsP , r , s .


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14

: ABC (. )

ABO, BCO CAO, cr2

1,

ar

2

1 br

2

1, rs.

2.1.7. a = 5, b = 7, = 30.

:

AB1C AB2 C, .

, c sin bh = 7/2. ,

sinsin

ba , . 30sin

7

5sin = 0,7.

1 = 44,427º 2 = 135,573º. 180, ,

1 = 105,573º 2 = 14,427º. ,

sinsin

ca c1 = AB1 = 9,63289 c2 = 2,49146. ,

, P1 2

1 hc

= 16,85756 P2 = 4,36006.

2.1.8. ?

: = 180 - - = 45.

sin2

sinsin2 b

P 34,5.

2.1.9. : ) 5, 5, 6; ) 5, 5, 8.

: 12, .

2.2.

. , , , , .


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2.2.1. n- (n = 3, 4, 5, …) a.

: , () n- R

r.

n

2 .

:n

aR

sin

2 ,

n

ar

ctg

2 .

n- n

naPn

ctg

4

2

.

4

32

3

aP 2

4 aP .

, 4

36

2

6

aP .

2.2.2. .) ђ . ) .) .

: )

72. ђ

36. ђ

108.

) 36ctg4

5 2

5

aP ,

510254

2

5 a

P 1,72048a2 .

) , d AD, AC BD. : AFD BCF.

xad , xaad :: , .

022 aadd ,

ad

2

5112 .

, .2

51:

ad .

. ( 1,61803).


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16

2.2.3. .

:

: 228

abP 2

2

22aaaa

, .

)21(2 2

8 aP 4,82843a2.

( 2.1.1.):

8 ctg2 28

aP ,

2122,5ctg .

2.3. Ч 2.3.1. d1 d2

sin2

214

ddP , ђ

.

:

AC = x + y BD = u + v .

ABCD ABE, BCE, CDE

DAE. )sin(2

sin2

)sin(2

sin2

4 vxyvuyxu

P ,

sin)(2

14 vxyvuyxuP sin)()(

2

1xyvyxu sin))((

2

1vuyx ,

.

2.3.2. . , ђ . .

2.3.3. . ,

180. , ,

.


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17

2.3.4. .

2.3.5. . .

, , .

2.3.6.

1. .

2. a, b h,

hbaP

2 .

3. a, b ababP sin .

3 . . , , . : k – ABC ( );r – (AQ);

AB – ; t – ;

– (AQB);

– (ACB);

– ђ (ABt);l – (AlB);AQBl – () ; ABl – () ;

ђ,

, . hdr , h .

3.1.

rs 2 .

2rP ,

r , = 3.14159... . ? ( r2 ).


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18

3.1.1. „“:

? , :

:

,

rlP , r . ,

l rl , 2rP .

rl , .

2

2

1rPI ,

sin2

1

2

1 22rrPPP IO ,

P ABQ ;

.

3.1.2. R2 n-.


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19

: ,...}5,4,3{n . a

P n- (

2a) l

P bP n-

( 2b), . :

bla , sin Ra , tg Rb , R

, n

n- . n . ,

tgsin RlR ,

cos

1

sin1

R

l. n

n

, 1cos . , sin Rl .

, 1R

, sin . ,nn

sin .

Rn

Rnn

Rnnl 22sin22 .

3.1.3. . 1 . ?

: rl 2 , r . RL 2 , R .

1 lL , 122 rR , . 2

1 rR 16 cm,

.

3.1.4. 13, 14 15 . .

: , 84

P cm2.

865

4

RabcR = 8,125.

2RPo 207,39; PPP o 123,39 cm2.

3.1.5. ђ . . .


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20

: r = a. , O1 O2 A B. O1O2A O1O2B

. AO1B = 1203

2 .

AO1B 3

2

2

1 2 aPI

,

AO1B 4

32aP ,

BO2A PPP Io

4

3

3

2 a 0,61418 a2.

2Po 1,22837 a2.

3.1.6.

. ђ . .

3.2.

(AB) (k). (t) (k ), .

(r) (O) (A). .

(AOC = )

(AOB = ) (AC) ђ

(ABt = ). ,

(180).

3.2.1. :


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21

) ; ) ; ) ;

) ђ ђ .

: ) , AOB, BOC COA

. COA

, . 1 = 2 2. 1 = 2 2,

= 2 . ) 180.) , .

= 1 + 2

ACD BCD . ) , .

.

3.2.2. , .

: PAD PBC

, (BD A + C =

180 BAD +DAP = 180)

PA PB = PC PD.

3.2.3. p = PA PB p = OP

2 – r

2.

P r O OP2 – r2.

4. () .

( ) , . . ()

, . . , . , (). .

, , . . .


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22

, . , . ( ) . ( ) . ,

().

, (), . . , , () , . () .

4.1.

1. ђ :) ; ) ; ) 50; ) 100;

( )?

2. :) ; ) ; ) 50; ) 100; , ?

3. , . .

4. () . . ?

5. () () A, p.

6. A B p. AC + BC .

4.2.

1. :I. a b c, ђ

, . a||c b||c a||b.II. a b ђ ,

, , . a||b a|| b|| .

III.

ђ , . || || || .


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IV. T, p, t , .

||p ||p T t t||p.

2. ,

. !

3. , , , . !

4. , , ђ . !

5. , . !

6. :I. T p t p.

II. A || .III. . IV.

.

4.3.

1. :I. ,

.

II.

( ) (1.I.).

2. , . !

3. , , .

4. , , , . .!

5. .

6. .

4.4.

1. .


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2. , , 12 13, 5 : 9. ђ ?

3. AB A’ψ’ A +1,

B +4. AB.

4. AB A’ψ’ [ A’ (+2), ψ’ (-3)]. AB.

5. ABC , A’ψ’ω’ AA’ = 1 cm, ψψ’ = 2 cm, ωω’ = 3 cm.

6. ABCD A’ψ’ω’D’ AA’ = 1 cm, ψψ’ = 2 cm, ωω’ = 3 cm. D.

5.

. , .

ђ .

.

.

, n д3, 4, 5, …жђ p1, p2, p3, … S p1Sp2 , p2Sp3 , p3Sp4 , … , n- .

n- ,

. , , ( ) . .

. , , .


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: , MNPQ ABCD MQ PN ABD BCD. , MQ PN

, MNPQ.

, , .

.

5.1.3. 1 2

(O) (p).

: , O A1A2, B1B2 C1C2, M, N P , . c1c2,

a1a2 b1b2, p = 1 2.

OA1A2 OB1B2 ђ M,

(c1c2). M

1 ( A1B1C1), 2 ( A2B2C2) OA1A2B1B2. , N P

1 2, . p.

, : M, N P p, .

M, N P . AB ( A1A2B1B2), BC CA , A1A2, B1B2 C1C2.

5.1.4. . ) ABC CDA . BD = a, ?


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) ABCD ADEF .

90, CAE ђ ?

: ) , DE ACD

2

3ah ,

BDE BEDhhha cos2 2222 , .

3

1cos BED , BED = 70,53.

. ) ,

2ad . AE, AC CE. ACE ,

= CAE = 60.

5.2. . . , .

5.2.1. !

: , ABC

Oxyz . p O P. p Ox,

Oy, Oz , ,

A1, B1, C1 .

p

Ox Oxyz, BC.

АА1 . , BB1 CC1 , P .

,

, . ,


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28

: „“, „“,„“, „“ , „ , , , “.

5.2.2. , , ђ .

, : A1OB1 > A1OB1.

, sin N1OB1 = 1

11

OB

NB

2

22

OB

NBsin N2OB2

N1OB1 > N2OB2, . A1OB1 > A1OB1.

5.2.3. .

: Oxyz , yOz, xOz xOy , .

+ > .

180, . , Ou

zOu = . A, B, C D x, y, z u OA = OB

= OD, „“ (COD COA)

ђ OD = OA. ABC, . BC + CA > AB BD = BC + CD> AB. OBD OAB , ,

. (4.2.2.) BOD > BOA, + > .

, , .

(5.2.3.) .

. , Sabc, aSc


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29

. SD ,

cSD = cSb. SB B SB =SD. B D S. C Sc, .

DC BC (BCS CDS). ABC |AC| < |AB| + |BCе, . |AD| + |DC| < |AB| + |BC|. |AD|

< |ABе ASD xOy.

yOz +zOv +vOx,


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vOx +xOt = vOt, tOy + yOz =tOz vOt + tOz +

zOv > xOy + yOz + zOv + vOx, .

5.3.3. 360.

: n > 3 (4.3.2.) n – 1 . , .

! .

Sabc c1 S Sc.

Sabc1 ( 4.2.3.) aSb < aSc1 + bSc1.

bSc + aSc , ,

, .aSb + bSc+ aSc < 360, .

ђ.


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31

II (: που – ; ο – , ) ђ . , , . . .

:

10 , 16 , 8 . :

22 , 45 , 25 .

: 12 , 30 , 12 .

: 20 , 48 , 24 .

(www.wikipedia.org ) (), ( ). : , , .

http://www.wikipedia.org/http://www.wikipedia.org/http://www.wikipedia.org/http://www.wikipedia.org/


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. : III

32

6 . . . :

ђ .

, .

. , .

. : , ,

. . () . , , a, b c. .

26aP 3aV

acbcabP 222

abcV MBP 2

BhV

( n-) (n ). .

6.1.

6.1.1. ( ) . ( ).


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. : III

33

: A C1 M BC.

6.1.2. ђ .

: x ђ AD1 DC1, . ђ. , ђ

AD1 , (DC1). AD1 || BC1 AD1 || BC1D, ђ, D1

(. BC1D).

D1O . OB DC1 E. ,

E DC1, BC1D

2a . BD1 3a .

.

BD1E 2

6aBE ,

2

21 aED

31 aBD .

,

D1BE , E.,

BEBD

EDBEBDBED

1

2

1

22

1

12

cos


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. : III

34

2

632

22

33

22

2

aa

aaa

23

4 . , = BEDBDOD 111 sin

3

3a .♦

6.1.3. DB1 BD1 ABCDA1B1C1D1 . ђ AC1 BD1.

: , DB1 BD1 S. BB1D1D , .

AB = a.

2aBD , 21 aBB .

aBDDB 211 .

S

, aAC 21 . SA = SB, ASB

. ASB = 60, .

ђ AC1 BD1.

6.1.4. AB1D1 BDC1 A1C ABCDA1B1C1D1 .

: , S , M N

A1C AB1D1 BDC1. , S , AB1D1 BDC1 . MS = SN. , A1S = SC A1M = NC. ђ, M AC1A1

SAMA 113

2 CA1

2

1

3

2 CA1

3

1 . NC MN ђ

A1C. , .

6.1.5. . .

6.2.

B, M P P = 2B + M.


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. : III

35

5.2.1. (, 2003.) 10 30 .

: ,

. 30 ,

4 / 230 10,6 . 10

30 .

6.2.2. (, , 1994.) 8 6, 13. .

6.2.3. (џ, , 1992.)

120. 16 14. , 20.

6.2.4. 32, 40. .

: MBP 2 )(2

1MPB 4)3240(

2

1 . ,

4, a = 2. aHM 4 ,

48

32

4 a

M

H .

6.2.5. 13 , 14 15 , 26 . . ) . ) .

6.2.6. 289, 17. .

6.3.

(1598 – 1647) . ђ . (Geometria indivisibilibus continuorum nova) 1635. , , ( 480 – 525.). (Arithmetica ifinitorum, 1655)


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. : III

36

, .

: ,

, , .

, a1 a2 , a p

F1 F2 ђ , .

:

, ,

, .

, 1 2 ,

F1 F2 ђ , .

(wikipedia.org) .

6.3.1.

. hBV .

: , h

, F1 F2,

1 = 2. ||

.

1 1 , . 1

= 2. , F1 F2.

= .


37/270

. : III

37

6.4.

5.4.1. (mathforum.org) . . 45 . ?

: n . n – 2 (n – 2)3 . , 45,

(n – 2)3 45. n – 2 3, . n 5. ,

45 , n3 45, . n 4. n 4 5.

n = 4. 23 = 8 45 – 8 = 37 . 43 – 23 = 56. 37 56 , 56 – 37 = 19 . ,

, , . 4

2 = 16 ,

19 . 16 16, , 12 . 19, n = 4.

, n = 5. , : 125, 27, 98, 18, 80. 25 , 80 , 4 . 5 , ,

9 , 27 45. 4 , , , . , 12 + 9 = 21 , 27 , 45. , 9 + 9 = 18 , 27 , 45.

, 5 4

, .

6.4.2. 4,

9, 10 17. .

: 9, 10 17,

a : b : c = 9 : 10 : 17. k a 9 , k b 10 k c 17 , k .

Bcsbsass ))()(( , s = 18k, . 48918 k k k k . 9

1k ,

9H , BHV = 36.


38/270

. : III

38

6.4.3. H = 20, d =21. :) ; ) ; ) .

: ) 204

3413 V 355,07; ) 4120 = 820; ) 2130,42.

6.4.4. D = 14,

641 d 372 d .

.

: 2222 Dcba , 2122 dca

2

2

22dcb 2

1

22dDb

2

2

22 dDa , b = 10, a = 7,

47c . bcacab 222

4734140 373,09. 4770 abcV 479,90 .

6.4.5. АА1, BB1 CC1 ABCA1B1C1 А0 , B0 C0. ABCA0B0C0 pAAAA 10 : , qBBBB 10 : rCCCC 10 : .

: H, B V , ABCA1B1C1, h BC ABC. , А0 BB1 CC1 B2 C2 ABCA0B0C0 ABCA0B2C2 B2B0C0C2A0. ,

hCCBB

AAPV ABC

23

1 020201

a

Pa

prHpqHpHP ABCABC

2

2

)()(

3

1

3

rqpHPABC

,

)(:3: 1 rqpVV . .

7 . (n = 3, 4, 5, …) ђ V (Vp1 , Vp2 , …, Vpn ) , n- . V , ,


39/270

. : III

39

. .

– . Ј

. .

, . .

, . . , . , . .

, , . n- .

n- n- . , ,.

. 4-

225 , 137 ( 147 ).

5150'. 2 583 283 6,25 . , ( - „ “).

7.1.

7.1.1. , : 1. ;

2.

; 3. ђ .


40/270

. : III

40

: T, A1,A2, A3, ..., An , S.

A1ST,A2ST, A3ST, …, AnST ( )

( ). A1S = A2S = A3S = … = AnS. , S , .

ђ .

7.1.2. . ( , 1965.)

. Vabcd ABCD. . ?

А Va .

А Vdc , AB || DC. , Vbc, AD || BC., ABCD . ,

.♦

7.1.3. 1. 2,

7 .

. .

2. a, 2a 2a. a H. .

3. 20, 21 . .


41/270

. : III

41

4. 10 24. 5 . .

7.2.

7.2.1. .

. ABCA1B1C1 . ( ), .

ABCC1,AC1A1B BB1C1A1 .

ABC, ђ

ACC1. AC1A1 . !

, ђ BC1C A. BB1C1 . . , !

, . ђ, . . , .♦

BHV3

1 , B

H . , h – , . .

, () , H h.


42/270

. : III

42

7.2.2. , , R.

.

.

AD, BD CD , DO ABC.

AOD H = R·tg . ABC

sin2Ra , sin2Rb .

),(sin2

1baabPABC

)sin(sinsin2 2 R ,

HPV ABC3

1 )sin(sinsin

3

2 3 R tg . ♦

7.2.3. ,

s. . .

. , ABCD a b. EF.

ABE

BCE cos2sa , cos2sb .

22 coscos sAF ,

22 cossin sEF .

,

223 cossincoscos3

4 sV .♦

7.2.4. H a. :


43/270

. : III

43

1. 22

10

55aHs

;

2. , .

22

20

525aHh

;

3.

5254

52545 22

Haaa

P ;

4. 5102512

1 2 HaV .

: ABCDE , a, , , R = OB r = F1O.

BF1 = a/2 BF1O OB 54њ,

54cos2

aR

2

ar tg 54°.

4

55236sin54cos

tg 54° = ctg 36° =

5

5255 ,

10

55 aR

5

525

2

ar , . aR 85065,0 ar 68819,0 .

OBF , 22 RHs ,

(1.) . F1FO 22

rHh , (2.) . ,

BCO , BCF. ,2

52

5 aharP

r h (3.) . ,

Har

V

25

3

151025

12

1 2 Ha .♦


44/270

. : III

44

7.2.5. a . ,

. a) . ) = 60њ .

. , AFG ,

, CDG . GG1 H , GG2 h , ABCDEF a.

,

() 2

3at

.

) G1G2G

G1 h. 321 aGG .

, 3aH tg , 32

3aV tg .

) = 60њ , tg 360 aH 3 . , h

sin

3ah , . ah 2 . a

31GGn BC je

a

t, .4

33an ,

.

BCG (

DEG), 221 nHh

194

31 ah .

, ABG ( EFG) 4

3

2

1 atm 222 mHh

4

372

ah .


45/270

. : III

45

2

22

222

21 ahahahaHM

.

2

36

2a

B , )19331910(4

2

a

P ,

2

14aP

. ♦

7.2.6. 1. ђ . 4, 8 9 m2.

[16 m3]

2. 45њ, d. .

[ 3

4

2d ]

3.

60њ . .

[ ,2

322 aa 12

23

a]

4. 5dm, 12/13 .

[25 dm2]5. 2 cm, 3 cm.

[ 284 cm2, 73

4 cm3]

6. 6 60°.

7. 3 ђ 45њ. 3. .

8. . 5 45њ 60њ. .

9. . b

30њ. .

8 . З

n- () . ђ n- .


46/270

. : III

46

(n- ) , , . ђ , .

,

.

8.1.

5 , 5 8 . , 8 (A, B, C, …, C1, D1 ), 6 12 .

, n- (n = 3, 4,5, …) n + 1 , n + 1 2n. ,

n- 2n, n + 2 3n .

. ,

() + () = () + 2.

8.1.1. ( ) a, a1 H. d, d1,

s, h () .

. 2ad , 211 ad . ACC1A1

212 )(

2

1ddHs . BCC1B1

2

1

2)(

4

1aaHh . MM1T1T tg

1

2

aa

H

.♦

8.1.2. a a1. s. .

. , BCC1 B1

212

1 )(4

1aasMM .

H T1T ђ .

TMM1T1 2

1

2 )(3

1aasH .♦


47/270

. : III

47

8.1.3. n . , .

. ,

2n . , 3n .

8.2.

8.2.1. 30 6, 5.

. , . P = B + B1 + M . . 7.1.1.

haa

aaP

2

4 1212 = 900 + 36 + 72h = 936 + 7213 = 1872.♦

8.2.2. 8 2, 5.

. , . 8.1.2.

haaaa

P2

34

3

4

3 12

1

2 = 31760 .♦

8.2.3. 1

. .

. ABCDE , A1B1C1D1E1 1, OGF = .

H = H2 – H1 =FO – FO1. h = GG1 BCC

1B

1. OG = r O

1G

1 = r

1.

BGO B1G1O1

2

ar tg 54°

2

1

1

ar tg 54њ ( 6.2.4.). OGG1O1

h = r sin , H = r tg , 2

11

aarrr

tg 54°.


48/270

. : III

48

, . P =

25

25 11

raar + h

aa

25 1

= sin)()(

4

5 21

22

1

2 aaaa tР 54њ.♦

8.2.4.

1. 7 3, 5. ?

[ 31 ]

2. 5 2, 2. ?

[ 13 ]

3. 10 4, 60њ. ?

[3]4. 5 3. 45њ. ?

[4 15 ]

5. a b, .

[ )(2 baab ]

6. 3 m, 8 m 2 m.

[ 3103 m2]

8.3.

, .

8.3.1. B1 B2, H,

22113

BBBBH

V .

: a1, b1, c1, ... a2, b2, c2,... H1 H2 T,

, 2121 :: aaHH 21 : bb 21 : cc

= ... , 22

2

121 :: HHBB ,

323

121 :: HHVV ,

)(:)(: 221121 HBHBVV .

.


49/270

. : III

49

(V) (V2)

(V1), . 12 VVV .

2

12 1

V

VVV =

3

2

3

122 13 H

HHB =

2

2

3

2

3

12

3 H

HHB . ,

))(( 22212

121

3

2

3

1 HHHHHHHH ,

HHH 21

.

1

3 2

1

2

1

2

12

H

H

H

HHBV

1

3 2

1

2

12

B

B

B

BHB 2211

3BBBB

H .♦

BBB 21 . BHV , . .

8.3.2. d1

ђ .

, H.

? , 361 d , = = 30°,

H = 2?

: sin2

1 211 dB .

M2

M3 111 NMd ,

M3M1M2 = .

M1M3 = H ctg ,

d2 = d1 – 2H ctg ,

sin2

1 222 dB .

22113

BBBBH

V . , 271 B , 322 d , 32 B 26V .

8.3.3. 3,6 m, 2,8 m 2,0 m. ,

= 2,5·102 kg/m3?


50/270

. : III

50

: B1 = 7,84 m2 B2 = 4,00 m

2,

V = 00,400,484,784,73

6,3 = 20,93 m

3.

= V· = 5232 kg.♦

8.3.4.

1. m:n?

[ 3nm

mH

]

2. 7 5 , 9 .

[109 m3]

3. 7 5 , 4 .

[654 m3]4. 7 5, 60њ.

[12

3109 m3]

5. 7 5 , 60њ.

[6

3109 m

3]

9 . ђ .

, , .

ђ, .

ђ, , „“ ABCDEF AEDG BCFG, , G .


51/270

. : III

51

. , .

. .

. , , .

, „“,

.

, „“. „“ .

9.1. .

: (). s i

1, i

2, ..., i

s , i

1 + i

2 + ...+ i

s.

ђ, . , .♦

9.2. S S = (t – 2)360њ, t .

: .


52/270

. : III

52

, , .

,

. . k , t – k ,

S = (k – 2)360° + (t – k)360° = (t – 2)360њ.♦

9.3. ( ) t , s , i : t + s = i + 2.

: s . i1 (), i2 , i3 , ... , s- is . () S = (i1 – 2)180° + (i2 – 2)180° + … + (is – 2)180°

= (i1 + i2 + … + is)180° – s360°= 2i180° – s360°,

i1 + i2 + … + is = 2i ( 9.1.).

( 9.2.) S = (t – 2)360њ, 2i·180° – s·360° = (t – 2)360° t + s = i + 2.♦

9.4. m , a n ( ),

tm = 2i, sn = 2i, 2mn = (2m + 2n – mn)·i,

t , s , i .

: (8.1.). n (), . n sn, s . sn = 2i, i .

, . m , . m ,


53/270

. : III

53

tm, t . , tm =2i.

, t = 2i/m s = 2i/n

( 8.3.) t + s = i + 2, 222

i

n

i

m

i

mnnm

mni

22

2.♦

, . , , .

ђ , ђ

ђ .

9.5. . :

1. () ,2. (),3. ( ) ,4. , 5. .

: 360њ ( 4.3.1.) n (n =

3, 4, 5, …) ( )

1802

n

n 360 m , m (m = 3, 4, 5, …)

.

3601802

n

nm nmmn 22

2

111

mn.

, n 3 m < 6 , m 3 n < 6. , :

n = m = 3; n = 3, m = 4; n = 3, m = 5; n = 4, m = 3; n = 5, m = 3.♦

9.6. (), (), (), (. ) () ().

1. 2. 3. 4. 5.


54/270

. : III

54

: n , m , t , s , i :

ns = 2i, mt = 2i, t + s = i + 2 2

111

mn

, :

1. n = 3, m = 3, t = 4, s = 4, i = 6 ,

. 2. n = 3, m = 4, t = 6, s = 8, i = 12

. 3. n = 3, m = 5, t = 12, s = 20, i = 30

. 4. n = 4, m = 3, t = 8, s = 6, i = 12

() , . 5. n = 5, m = 3, t = 20, s = 12, i = 30

.♦

()

. (t + s = i + 2) (t, s) , .

, . , . .

, ђ (, 350. ....). , , , , .

: (4), (6), (8), (12) (20).


55/270

. : III

55

9.7. ?

: , . , .

s , r . As(a), Ps. Vs. , :

Ps = s·As(a), )(3 aA

sr

V ss

sP

r

3 .

, :

324 aP ,2

6 6aP , 32 2

8 aP ,2

12 )525(53 aP , 35 2

20 aP ;

26

3

4

aV , 36 aV ,

3

23

8

aV , 312

4

575aV

320

12

)53(5aV

.♦

9.8.

1. .

[ ]2. : . ? . ? . ?

[3, 6, 7]


56/270

. : III

56

3. , ABCD a.

a. AD BC ?. DT ?

. ? . r R ? . DT ?

[2

2a,

3

6a,

12

23a,

12

6a

4

6a, 3:1]

4. ( )ABCDA1B1C1D1, .

a. ђ

( AB1, AD1).. ђ

( AB1,BC1).

[60°, 60°]5. . ђ .

[3

a, . 6.1.2.]

6. . [ ]7. .

. . . .. . . .

[ 32 2a ,3

23

a,

6

6a,

2

2a]

7. . 2510253 a 20,64572 a2.

8. .

35 2a 8,66025 2.


57/270


58/270

. : III

58

, ђ , ђ , .

, . g s (360°). s . , AB s, , , .

.

10 . В

(), , ,

. . . , () () r, s H. .

( .

) , . ,

. ( ) , .


59/270

. : III

59

10.1.

(T) (r1 + r2) (F1 F2) . , ,

(AB) () . , .

, F1 F2 F1 F2.ђ .

10.1.1.

, .

(. . , 1822.). . , F1 F2. k 1 k 2.

F1T TM1 , ђ F2T TM2. F1

F2 M1M2 k 1 k 2. , ,

F1 F2 , .♦

10.1.2. r = 4 H = 6. . .

: , AB = a, DE = a. EO x. CDE AOE

222

)2( Hxa

2

22

2

a

rx ,

, .

2

4 22 rHa

, 25a 7,07.♦


60/270

. : III

60

10.1.3. r . .

: 2 r. ,

H = 2r. . , a1 = A1B1, a2 =

A2B2. ABO ra 1

3

322

ra . raM 26

12 MMM )(12 12 aar )332(4 2

r .♦

„“ „“

(2r ),

. , .

MBP 2 ,

rHrP 22 2 )(2 Hrr ,

3,14159… .

,

, . , .

10.1.4. 3 4. r H .

: ABCD AC = 4 BD = 3


61/270

. : III

61

. AFC 3, 4

5, 522 r . ,

2

5

2

43 H

.

4

5r

5

12H .♦

(, ) (

) , . HBV ,

HrV 2 .

10.1.5. H = 10 cm

d = 5 cm. = 2 /3. .

: MN , ,

P1 P2. MNO je sin2

1 2rP MNO .

,

2

2

1rPi . MNOi PPP 1

)sin(2

2

1 r

P .

2sin2 dr 2,89 cm,

2sin

sin

8 2

2

1

dP 334

36

2 d 220,0 d .

HPV 11 , . V1 51 cm3.

HrV 2 262 cm3, V2 = V – V1 211

cm3.♦

10.2.

9.2.1. ?

: HrM 2 , H r

222 rB .

222 rHr r = H. .♦


62/270

. : III

62

10.2.2. H = 8 , , R = 7 r = 6.

: rR MMBP 2 ,

rR BBB )( 22 rR ,

HrRMM rR )(2 .

,

))((2 HrRrRP , . P = 234 .♦

10.2.3. ,

. .

: ABCDS. , 2r = H,

ABS 2

3ah .

)12( aH ,

: P = 6r2π = 2

2)223(3 a .♦

10.2.4.

1. P0 = 120 cm2, B = 78,54

cm2. d = 3

cm . [96,00 cm2]

2. r H . , .

[4

2

2

Hr ]

3. P0. ?

[4

0 P

]

4. ђ r H , .


63/270

. : III

63

[r = H]5. . .

[ : 3]6. .

[ )21(2

a ]7. (

) 80 cm2.

[120 cm2]8. H

. .

[H2·tg ]

9. () 10 .

() 144 cm2. . [4 cm, 18 cm]

10. ,

12 cm2, 5 cm.

[ 324 cm2, 39 cm3; 38

321

cm2,

4

327 cm3]

10.3.

H B V = BH.

10.3.1. r = 564 l = 10 . ?

: hrV 2 .

2r

Vh .

r = 5,64 , 10 10 dm3

h = 0,10 dm.♦

10.3.2. r = 6 cm 7 9 . ?

: . , , . ,


64/270

. : III

64

, .

2

21 ss . ,

,

, s1 + s2 . ,

2

212 ssrV , . V = 288 cm3.♦

10.3.3. V.

: BHV Ha

4

32 ,

.

6

3ar .

HBV vv Hr 2

H

a

4

3

33

1 2V

33

1 .♦

10.3.4.

1. .

[V1 : V2 = r1 : r2]2. .

[M1 : M2 = r2 : r1]3. .

[4

3 a]

4. .

[4

3 3 a]

5. V.

.

[ V3

34 ]

6. S . l. .

[Sl]


65/270

. : III

65

7. , . .

[4:1]8. . ?

[8

2

3

a ]

11. (, ) . , SPO O OS h, PO r SP s, OS. S, .

SP = s, . P

2r , r = OP = 3,14159... . S OSP . P S s, . .

s. , .

, 2

2

1sM

l = P1P2 = s . ,

l = 2r , s

r

2 .

.

11.. M = rs .♦

11.. ()

() , :

2sin2

.

: r,


66/270

. : III

66

s, .

,

OPS

/2.

s

r

2sin

.

l = 2r .

, s

. P1P2 l = s . ђ

2r = s , .

2

s

r,

2sin2

.♦

, 60њ ,

3

= . , ( = 90°)

() 30њ.

, ,. P = B + M.

11.. P = r (r + s).

: B = r2 , M = rs , P = r 2 +rs , .♦

S . .

(H), , .

HrV

2

3

1

. .

11.. :i. ; ii. .


67/270


68/270

. : III

68

. ( 262. – 190. ....) , , , . , , ,

.

11.1.1. , (). , (). .

, (). ,

, .

, (). , , .

, . , , .♦

11.1.2.

, . (P) (F1 F2) :

PF 1 + PF 2 = const.

F1 F2.

(P) (F)

(d) .

A F ABC BC d.


69/270

. : III

69

(P) (F1 F2) :

PF 1 – PF 2 = const. F2 A,

F1.♦

- . . (1794 – 1847. .) , , . , .

11.1.3. (), .

: , S G1 G2, F1 F2. k 1 k 2. P P1 P2 k 1 k 2.

k 1 k 2 , P

1P

2

P.

, PF1 P1P (G1) ђ , PF 2 PP2 (G2). PF 1 + PF2 = P1P + PP2 =P1P2 = const. ♦

(, , , ,

) . , , .

11.1.4. d F.

P PF Pd . F (), d, . 0 < < 1


70/270

. : III

70

(. ἔ ιι – ), = 1 (. πααβοή - ), > 1 (. ύπβοή – ).

(10.1.3.) ђ , (Xah Lee

Web, http://xahlee.org/).

11.1.5. 1. .

, . .

2. . 3.

.


71/270


72/270

. : III

72

[ , . .

, .]

11.2.

() () . „“ „ “. ,

M = rs ,

B = r2 ,

P = r (r + s).

, s2 = h2 + r2.

() (S) , .

() ,

2sin2

.

11.2.1. H, 180њ.


73/270

. : III

73

: , = , 2

1

2sin

3

.

ђ 6

, . 30°. ,3

32Hs

3

3Hr , P =

H2 .♦

11.2.2. a b a. .

: ABCD AB = a AD = b ђ

BAD = AB. D C r D1 C1. M1 M2.

M1 = rb M2 = 2ra b, a

r = b·sin Pa = 2 a(a + b)sin .♦

11.2.3.

1. 100њ . .

[32°]2. H. , 120њ.

[2

2

H ]

3. 1 , ½ .

[90 cm]

4. : H. ?

[

cos1

2

H]

5. a = 10, b = 17 c = 21 :

a. ; b. ; c. . . [

a

cbPMa

)(2 = 2553,6· ; Mb =1225,4· ; Mc = 864· cm]

6. 11 cm 3 cm 105 cm . ?

[570 ; 397,8 cm2]


74/270

. : III

74

7. . 24 cm 18 cm, 7 cm. .

[344,9 cm2]

11.3.

, . , 4-

. .

( || )

, . ђ . ,

(5.3.) .

, , , 1 2 O k

(X1 1) (X2 2) OX2 = k · OX1.

O , k .

. A1B1 A2B2

A2B2 = k · A1B1. .

, .

BHV3

1 .

. ђ, , . ђ, r :

HrV 2

3

1,

H ( ) .


75/270

. : III

75

11.3.1. 384 cm2, 20 МЦ.

?

: 384 = r · (r + s), s. r

2 + 20r – 384 = 0.

r = 12 cm. s2 = h2 + r2, h = 16 cm.

16123

1 2 V cm3 = 768 cm3.♦

11.3.2. 13, 14 15 . .

: 13 15

, . r 14 cm . ђ 14 cm.

P = r·13 + r·15 =

r·28 . yrxrV 22

3

1

3

1 )(

3

1 2 yxr

143

1 2 r .

(r)

))()(( csbsassP

, s =

21 . , P = 84 cm2,

P

r

2

14, . r = 12 cm. P = 336 cm

2 V = 672 cm

3.♦

11.3.3. H = 10 = 0,77 . .

: 0 = 1 ,

.

HRV 2

3

1 , m = V HR

2

3 ().


76/270

. : III

76

hrV 203

1 .

0r2h = R2H. , r : h =

R : H (. ) RH

hr ,

3 Hh 9,17 .♦

11.3.4. 1. r,

. ?

[ 14

3 2

23

r]

2. s 60°. ?

[24

33 s

]

3. a, b, c . .

[cba

1:

1:

1]

4. 7 cm 5 cm 2 cm , . .

[17 cm3, 21 cm3]5. .

. [1 : 4]

6. , r . . .

[9 : 2]

7. 0 = 0,89 g/cm3 .

, 90% ? [0,65 g/cm

3]

8. H ( ) . ?

[ 3 11 H ]

12 . З

ђ .


77/270

. : III

77

: h – ,R, r – ,s – M – .

22 )( rRhs ,

M = s (R + r),

P = R2 + r2 + s (R + r),

hrRrRV )(3

1 22 .

12.1.

.

12.1.1. :

P = R2 + r

2 + s (R + r).

: . S ,s . , H , h . ђ, R , r

, H – h, S – s.

, S s (R – r) : R = s : S.

M = RS – r(S – s) , M = s (R + r).

, R2 r2 , , .

P = R2 + r

2 + s (R + r).♦

12.1.2. .

M = s (R + r).

:


78/270

. : III

78

. , S S – s

, ,

S = 2 R (S – s) = 2 r.

RS 2

rsS 2 ,

)(2 rRs .

, ђ

22 )(2

1

2

1sSSM . )(

)(2 2rR

rRM

,

M = s (R + r). , (11.1.1.).♦

12.1.3. ( R, r, h)

? , , ( 7, 5, 6)?

: )(3

22 rRrRh

V

.

4

)( 2 rRhVv

. vVVV ,

2)(

12rR

hV

. .

vV

VV

.

, 218V , 216vV .

2V ,

216

2V = 0,009259… , . 1%. ♦

12.1.4. d1 = 15 cm, d2 = 8 cm . .

: ABCD . ђ,

AC = d1 BD = d2 . , p A BD. AC AC/2,


79/270

. : III

79

AC/2.

, . P = 2· (AC + AC/2)·CD +

2· (AC/2)·CD = 4 ·AC·CD = 222

112 ddd .

, .V =

2·(1/3) (AC2

+ AC·(AC/2) + (AC/2)2

)·(BD/2) – 2· (AC/2)2

·(BD/2) = 22

1

2 dd

. ,

P = 510 cm2 V = 900 cm3.♦

12.1.5.

1. r. . .

[

1

sin

42

2

2

r ]

2. p = 24 cm, = 10 cm

. . [240 cm

2]

3. . .

[6a2 ]

4. R = 25, r = 15, s = 10 ? [25 : 9]

5. R H . .

[4 : 1]

6. P0 = 15 cm2, R = 7 cm r = 3 cm. ?

[108 cm2]

12.2.

, , .

12.2.1. B1 B2, h,

2211

3 BBBB

h

V .

7.3.1.♦


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. : III

80

12.2.2. hrV 2

3

1 (r h

) .

: n (n N ) n – 1 ђ . k- ( k =

1, 2, …, n) n

h,

n

k r

n

k r

1 .

k-

n

hAH ,

n

k rAD 2

n

k rGF 1

2

. Vk

AD GF. k

n

k

n

k rV

n

k

n

k r k

221

,

2

2

32)1( k

hr

nVk k

.

k = 1, 2, …, n

222

2

3222 ...21)1(...10 n

hr

nVn

,

V = V1 + V2 + … + Vn . n – 1 n

6

)12)(1(

6

)12()1(2

3

nnn

hr

nV

nnn

.

n3 ђ

3

2

11

11

3

2

11

11

2

nn

hr

Vnn

.

, n. n ,

1/3. , n 31

2

hrV

,

. hrV 2

3

.♦


81/270


82/270

. : III

82

[ 722,3 cm3]

13. R ,

. ().

, .

24RP ,

R .

. . .

3

3

4RV .

. , R . r. , h

)3(3

2

hRh

V

.

() , .

RhP 2 ,

R h .

, . , , , . , .

hRV 2

3

2 ,

R , h .

, ђ ,


83/270

. : III

83

.

RhV 2 , R , h .

ђ

, ђ , , r1 r2.

)33(6

22

2

2

1 hrrh

V

,

h , r1 r2 , .

13.1.

() R

24RPl ,

3,14159. ?

12.1.1. , . () R () ,

( 2R )

(2R). ( ) RRPc 22 .

.

: , ђ , .

, џ.

, , .ђ, , ђ

ђ .


84/270

. : III

84

, hc , hs . џ

2R hc ,

2r hs. ,

r, .

, K . PA P, a A QP B. APK

. KPQ , QKP + KPQ =

90. ђ KPQ + BPA = 90, . QKP = BPA.

, : QKP

BPA. PQ : KP = AB : PA, PQ = r, KP

= R, AB = hc PA = hs, . r hs = R hc.

r hs џ

.

Pk = 2R h, R h ,

Pz = 2R h, h .

13.1.2. R.

. .

: )(3

2

221

2

1 rrrrH

Vk

, H

, r1 r2 .

H = 2R, r1 = R ctg2

r2 = R tg

2

,

)

2

tg1

2

ctg(

3

2 223 RVk .

13.1.3.

1. . .


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. : III

85

[12

6ar ,

4

6aR ]

2. . .

[ 6

6a

r , 2

2a

R ]

3. h = 2,

= 3:2. ?

[ )41)(1( 22 h = 100 ]

4. . ?

[1/5]

5. 400 6, 8. .

[40 , 280 ]

6. r = 6 cm d = 12 cm.

[36π cm2]

13.2.

13.2.1.

3

3

4rV ,

r .

: 3

3

4rV ,

(6.3.) . r, , ()

ђ r, (1).

(2), h () , k h , ph r - h r. .

222 )( hrrx = 22 hrh ,

(k h) )2( 2hrhPk . (ph)


86/270


87/270


88/270

. : III

88

: n N OA0A1, OA1A2, OA2A3, …, OAn-1An , ђ r1= r2 = r3 = … = rn = r , s , r n .

,

1

1

13P

rV , 2

2

23P

rV , 3

33

3P

rV , ... , n

n

n Pr

V3

,

nVVVV ...321

)...(3

321 nPPPPr

rPr

3 ,

Pr = 4r2π r.

243

rrV = 33

4 r .

13.2.6. , 7, 15 cm , 25 cm. .

: , . . .

h1 = C1C2 = 15 – 7 = 8 cm, OB1C1

OB2C2 22

1 1525 r = )1525)(1525( = 4010 = 1004 = 210 = 20,

22

2 725 r = 3218 = 649 = 38 = 24.

)33(6

22

2

2

11 hrrh

V

=3

11968 cm3.

h2 = 15 + 7 = 22 cm

3

375322

V cm3.


89/270


90/270

. : III

90

2. . . s, , , . :

i. , ;

ii. ; iii. .

3. r h. .

4. r1 r2.

.

5. . . .

6. 1 : 3. ?

7. 3 cm 108π cm2. ?

8. 6 dm 216π dm3. ?

9. 1 m , 14 cm 30 cm. :

i. ; ii. ђ .

10. √

1.

11.

dm

1 dm.

12. .


91/270

. : III

91

IV . , , , .

m n

1211 ),...,,( bxxxf n

2212 ),...,,( bxxxf n

...

mnm bxxxf ),...,,( 21

f i x j n- (i = 1, 2, …, m; j = 1, 2, …,n) x1, x2, …, xn, f i bi . , f i x j.

, , . , :

1. ;2. ;3. .

, , .

14. .

f i x j :

n

j

jiji xaf 1

, i = 1, 2, …, m.

, f 2 = 3x1 – 2x2 + x3, x1 , x2 x3.

14.1.

ђ „ “ (Zhou Bi

Suan Jing:九章算术), 10 – 2. ... 14.1.1.

; 35 94. , ?

( ), 70, 94. , 24 – . 24 : 2 = 12, 35 – 12 = 23.


92/270

. : III

92

, – , – , :

x + y = 35

2x + 4y = 94

= 23, = 12.

„ “, VII, , ђ .

14.1.2.

9 11 . , . ђ,

ђ , , 9 (8 1 ) 16 (). ?

, – , – ,

9x = 11y8x + y + 16 = 10y + x

: x = 44, y = 36 .

, , ( ) , ђ, .

, 1 = 55 (). 1 = 9 1/11 = 45 (). , ()

8x1+ y1 + 16 = 10y1 + x1.

ђ, 855 + 45 + 16 + 501, 1045 + 55 +

505, 4 . k 1 = 4 „ “, „ “.

Д , 2 = 22 . 2 = 9 2 /11 = 18. ,

8x2+ y2 + 16 = 10y2 + x2.


93/270

. : III

93

, 822 + 18 + 16 = 210, 1018+ 22 = 202. , 8 . k 2 = 8 „ “, „ “.

21

21

xx

k k

,

21

2211

k k

xk xk x

=

84

558224

= 44 .

ђ „“, . x2 < x < x1. ђ y = 9x/11 = 944/11 = 36 .

, , 2 12 , 2 4 (1 = 16 ).

, 18. .

, , .

14.1.3.

, 360 .

, . ?

x – , y – , z – .

x + y + z = 360 y = 2z x = 2(y + z).

, x = 6z, 9z = 360, z = 40. x = 240, y = 80 z = 40. , ,

240, 80 40 .

14.1.4.

6 , 4 ђ, ђ. ?

x1, …, x6 ђ.


94/270

. : III

94

(( 41 j j xx ) ( j д1, …, 5})) ( 61 3 xx ).

,

( 465

xx , 864

xx , 1263

xx , 1662

xx , 2061

xx ) (61

3 xx ).

20366

xx , . x6 = 10.

, ђ 10 , 14, 18, 22, 26 30

.

14.1.5.

1. . 6 , 8 . 12 86 ,

? [7]2. $1.5 $4. 2200 $5050. ?

[700]3. 7. , 27. ?

4. ђ џ. џ 13 4 487£. џ 6 232£. ,

?

5. 1800 (. jetstream). , , 4 . ?

: v – w – . 3(v + w) = 4(v – w) = 1800.

6. (–1, 9), (1, 5) (2, 12).

: cbxaxy 2 .

7. :

22

7523

xxx

x.


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. : III

95

: 112

x

C

x

B

x

A.

.

8. 100 , 3 ( 1, 2 3 ). $100. 50 , 3 , 10 , 100 100 ?

: - 100, ђ, 1 2 3 100. 2 = 20 + (19/5) 3, .

14.2.

m n

11212111 ... bxaxaxa nn

22222121

... bxaxaxann

…

mnmnmm bxaxaxa ...2211 .

aij (i = 1, 2, …, m; j = 1, 2, …, n) () , bi , x j , . (m = n) , , .

mmnmm

n

n

yaaa

yaaa

yaaa

...

...

...

...

21

222221

111211

n β 1, β 2, ..., β n, (x j = β j) .

: , ( ),

() . , ђ, ђ. :

ђ ђ

2

1

21

21

xx

xx

1

1

21

21

xx

xx

1

1

21

21

xx

xx


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. : III

96

, .ђ ђ . , .

, XOY. . , , ђ , ђ .

: 336 yx 224 yx ,

336 yx 224 yx , .

3- .

(y1 = 0, y2 = 0, ..., ym = 0) , . , , (x1 = x2 = ... = xn = 0),

. ђ .

, , (Carl Friedrich Gauss, 1777 – 1855), - , . .

mnmn

nn

nn

bxa

bxaxa

bxaxaxa

''

...

''...'

''...''

22222

11212111

, (m-) mnmm abx ' / ' .

xn-1, , , , , .


97/270


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. : III

98

, -1 . :

1200

0000

2

3

321

x

xxx

x3 = ½, -. , x3 x1 + x2 = 3/2. , x2 = t, ђ x1 = 1,5 – t. , ђ, () ђ (x1, x2, x3) = (1

– t, t, 3/2), t .

14.2.4.

1. 2- :) ) )

864 yx

1096 yx

864 yx

1269 yx

1269 yx

846 yx

2. :) )

3

2

1

321

321

321

xxx

xxx

xxx

3

1

2

321

321

321

xxx

xxx

xxx

[(2,5; -0,5; -1) (-0,5;1,5;1)]

3. :) ) )

12

1

0

321

32

31

xxx

xx

xx

12

1

0

321

32

31

xxx

xx

xx

22

1

0

321

32

31

xxx

xx

xx

[(0,5; 0,5; -0,5), , (-t; 1+t; t)]

4. :) ) )

2x1 + x2 + 3x3 = 1

2 x1 + 6x2 + 8x3 = 36 x1 + 8x2 + 18x3 = 5

3x + y – 6z = – 10

2x + y – 5z = – 86x – 3y + 3z = 0

x + z = 1

x + y + z = 2x – y + z = 1

[( 3/10, 2/5, 0); (t – 2, 3t – 4, t); ]

5. :) )


99/270


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. : III

101

)32()2(2

)32(2)2(3

21212

21211

xxxxx

xxxxx

ђ

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