( )

()

-

..

-

2009

( )

()

-

..

,

,

-

2009

87(03)

.. :

/ . .-, 2009.

, -

, ,

.

, .

.

, -

.

.

. 38, . 6, . 7 .

:

.., . . .

.. , . . .

, 2009

......

1. ...

2.

2.1. ...

2.2.

2.3.

3. .

3.1.

.

3.2.

3.3.

4. ...

4.1. .

4.2.

4.3.

5.

6.

.

4

6

19

24

31

43

54

57

59

63

67

69

76

81

85

94

99

102

1.

.

. -

.

. , -

, , -

, .

, -

, -

.

0 360 -

.

, .

, -

-

. , ,

.

-

-

.

,

. -

- ( ),

, .

-

: ,

(.1).

-

, ,

. ,

, .

, -

, .

, ,

.

.

. 1. (), () ()

, -

, . -

( +, -, , -

() , -

,

() , ).

, -

, . -

( +, -).

, -

, . -

. -

= + . , -

, . , -

, .

,

. , -

. , , ,

, -

, () .

: -

( ),

(

).

, , -

, , ..

. -

(.2). ,

. ,

, .

. 2.

, :

= + ;

= - ;

= + = + + ;

= - = - - ;

= + ;

= - ;

= - ;

= - ;

= - ;

= + = + . -

, -

. .

1. = 86; = - 3; = 9. , .

. -

, ( ).

= + = 86 + (- 3) = 83;

= + + = 86 + (- 3) + (+ 9) = 92;

= + = (- 3) + (+ 9) = + 6.

+

-

2. = 264; = + 2; = - 14. , .

. , ,

:

= - = 264 2 = 262.

, , , -

, .

= + = 262 + (- 14) = 248.

= - = - 14 - 2 = - 16. 1. = 236; = + 5. .

2. = 22; = - 2; = - 8. , .

3. = 164; = + 2; = + 12. , , .

4. = 328; = - 4; = + 6. , , .

5. = 78; = + 6. .

6. = 8; = - 4; = - 9. , .

7. = 285; = + 5. .

8. = 136; = - 4; = - 13. , , .

9. = 44; = + 3; = - 7. , .

10. = 155; = - 11. .

11. = 268; = - 4; = - 13. , .

12. = 331; = + 2; = + 5. , , .

13. = 16; = - 4; = + 1. , , .

14. = 69; = - 5; = - 15. , .

15. = 96; = + 3; = + 7. , , .

, ,

, .

-

. .

-

, -

.

() , -

, , -

() .

,

, -

. -

.

, -

, -

.

, -

, .

:

= + ;

= + ;

= - .

, (

, ).

(, ), -

.

, , ,

= - .

( = ), ,

( ) , .

( ) -

() :

= .

,

( ), -

, .

. - . ,

-

(. 3). C

1 2

S S

. 3.

( ), , -

, . -

, ,

:

= - .

,

, , , , , -

, . -

, -

:

= + .

,

, .

. -

, . ,

: , .

, .

S, -

(. 3),

1, :

= S tg .

-

(. 4): 4 5 S

. 4.

= arctg (/S).

= /S

= 57,3 / S 60 /S,

, , ,

S/60.

, ,

: () 60

1

.

.

60 , , ,

, 1 , 3 . -

1, , , , 60 ( 120 ),

.

3. = 4, S = 150 . .

. 2,5 , 60 ,

, . = 2,54 = 10 .

2 -

(. 5).

1 2

S (t) S (t)

. 5.

S, -

(. 5), 2,

, -

:

= S tg ;

= arctg (/S);

60 /S;

S/60.

( 60 ).

, , , -

, , .

, , -

:

S/60 S/60.

-

S S,

/ S = / S.

,

/ t = / t.

, :

= (S/S)

= (t/t) .

, -

, (), -

. ,

:

= + .

,

. ,

, , -

, . -

(. . 5),

.

.

4. = 236; = - 3; = +9. .

. 1. :

= + () = 236 + (+ 9) = 245.

2. :

= + () = 245 + (- 3) = 242.

5. = 312.

, = 317; = + 8.

. 1. :

= 312 + (+ 8) = 320.

2. :

= 320 - 317 = 3.

6. = 82.

= 74; S = 72 ; = - 4 . .

. 1. :

= 60 (- 4) / 72 = - 3.

.

60 , 4 , 4.

(72 ), (3).

, .

2. :

= = 74 82 = - 8.

,

. 74, -

82, 8 ,

. , , -

8 , - 8.

3. :

= + = (- 8) + (- 3) = - 11.

4. :

= 82 + (- 11) = 71.

16. ,

= 90. = - 6.

- 9. .

17. ,

= 210. ,

= 202. , = - 8.

18. ,

= 356.

= + 6. ,

= 5.

19. = 243.

= 248.

S = 45 = 6 . .

20. 345.

= 340, = 338.

.

21. , = 33.

,

28, + 7,

= 30.

22. ,

= 268. = 280.

, = + 6.

23. = 137, -

= 130. 75

4 .

.

24. = 27

= 24. , -

S = 90 = 6 .

25. = 58.

65. , ,

, 4.

2.

, .

(.6) :

- , -

;

- , -

;

- , . -

( ).

. -

, .

-

. . -

-

, , .

, , , -

, -

- .

-

760 . . (1013,2 ) .

-

760 . . (1013,2

) , -

.

. 6. , -

.

- -

, ,

. -

, , .

-

-

, . ,

. -

.

-

= + -

.

. .

760 ..

760

- -

,

.

-

.

+ 15 - + 15. -

;

. -

0

tH

300

15tH

= ,

t0 ;

H .

-10

. 7, 8 9

(.7). t0 + tH H 7 8 9 + H

. 7.

t0

:

.t 11)( ++= , ;

;

t ;

,

;

. ,

;

.

, , -

, , -

:

11

. += .

:

11)(H t. ++= . 7. , -

= 7270 . = - 70 , = 476 ,

t0 = - 8, t = - 42, . = 744 . .

.

. 1. :

= + = 7270 + (- 70) = 7200 .

2. :

t0 + t = (- 8) + (- 42) = - 50 .

3. -10 :

= 6780 .

4. :

= (760 744) 11 = 176 .

5. :

= 6780 176 476 = 6128 .

26. = 166 .

1500 . ,

318 .

27. = 294 . -

1300 . ,

-

88 .

28. ,

68 , -

900 .

29. 8100 . -

- 90 . ,

= 263 , t0 = + 8 , t = - 32 , = 734 . .

30. 6600 .

350 , 80 , -

t0 = + 12 , t = - 42 .

.

31. 10200 .

- 100 . t0 = + 5 ,

t = - 42 . - 310 . -

.

2.1. ,

( )

.

,

.

,

, ,

.

-

: (

);

; -

; -

(. 8).

. 8.

.760

.

760 ..

-

, , (. 1).

1

. ,

, /

300 > 300 300

100 200

-

300 301... 550

> 550 600

100 200

-

( 2000 ) 550 > 550 900

300 -

, , -

( 2000 ) 550 > 550 900

600 -

:

)

t. += . , 100.

, .

,

. 1 H.=300 ,

. H

, ;

)

t 300 += . , 10. -

100

, . -

( 4) -

.

.

-

, .

H -

50 -

, .

)

11)(760 .t.. ++= . , 10. -

. 760 . -

( -

).

) 760 . .

11)(760 .t.760. ++= . .760

. , -

.

)

t.. += ; )

t.. += ,

. , ; -

, ;

t ,

- , ,

, , (

), ;

, ;

300

/, .

. -

, , . .;

.

, , . .

8. , -

t0 = + 8 ,

= 86 .

. 1.

:

. = 300 .

2. :

= . + (t0 15)( . + )/300 = 300 + 86 (8 15)

(300 + 86)/300 = 495 .

:

= 500 .

9. 760 .

. , = 246, = 568 , t0 = - 24 ,

. = 754 . .

. 1.

:

. = 600 .

2. :

. = . + = 600 + 568 = 1168 .

3. . :

t = t0 6,5 ( . , ) = - 24 6,51,2 = - 32 .

4. :

t0 + t = - 24 + (- 32) = - 56.

5. :

. . = 1350 .

6. :

= (760 - . ) 11= (760 754) 11 = 66 .

7. 760 . .

.760 = 1350 + 66 = 1411 .

8. .

.760 = 1800 .

10.

, t0 = - 12 , = 47 , V = 240 /.

. 1.

:

. = 100 .

2. :

= . + = 100 + 47 = 147 .

3. :

0

t H30015t

H = = 1473001512

= - 13 .

4. :

. = 147 (- 13) = 160 .

11.

, t0 = + 22 , = 263 , V = 160 /, . = 746

. .

. 1.

:

. = 100 .

2. :

= . + = 100 + 263 = 363 .

3. :

0

t H30015t

H = = 3633001522

= 8 .

4. :

. = 363 8 = 345 .

32. -

, t0 = +16 , = 138 , V = 320 /.

33. -

, t0 = - 18 , = 254 .

34. , t0 = + 4 ,

= 98 , V = 160 /.

35. , t0 = - 2 ,

= 24 , V = 240 /.

36. , t0 = + 8 ,

= 324 , V = 330 /.

37. , t0 = + 12 ,

= 446 , V = 400 /.

38. -

, t0 = - 26 , = 228 , V = 260 /.

39. -

, t0 = + 16 , = 689 , V = 220 /.

40. -

, t0 = - 24 , = 294 , V = 340 /.

41. -

, t0 = - 24 , = 2018 , V = 240 /.

42. ,

t0 = - 18 , = 64 , = 743 . ., = 218 , V = 390 /.

43. -

, t0 = +28 , = 248 , = 743 . ., = 781 ,

V = 430 /.

44.

760 . .,

t0 = + 11 , . = 753 . ., = 217 .

45.

760 . .,

t0 = - 39 , . = 738 . ., = 461 .

46.

760 . ., t0 = - 26 , . = 746 . ., = 2413 .

47. 760 . .

, -

= 24, = 268 , t0 = - 24, . = 758 . .

48. 760 . .

, -

= 342, = 627 , t0 = + 24, . = 747 . .

49. ,

= 149, = 1568 , t0 = - 24,

. = 742 . .

50. , = 296,

= 2255 , t0 = - 42, . = 734 . .

2.2.

()

-

,

.

, ,

() (. 9).

-

760 . . (1013,2 )

.

-

.

. 9.

S

= 760 .. .

.

t

-

:

1. . :

. = (760 )11,

, . .;

2. :

= . ,

, ;

3. t:

t = /V,

V , /;

(. 10): t V 1 1 2 2 10 V 10 t

. 10. ()

4. :

= + t,

;

5. ,

S:

S = W t,

W .

(. 11): S W 1 2 t

. 11.

12. = 9600 ; -

= 600 ; = 746 . .;

W = 640 /; -

V = 11 /; -

= 18.15. ,

.

. 1. :

. = (760 746)11 = 154 .

2. :

= 9600 600 154 = 8846 .

3. :

t = 8846/11 = 804 = 13,4 .

4. :

= 18.15 + 0.13,4 = 18.28,4.

5. , :

S = (640 13,4)/ 60 = 143 .

51. = 7200 ;

= 400 ; = 766 . .; -

W = 410 /;

V = 6 /;

= 10.32. , -

.

52. = 4500 ;

= 500 ; = 716 . .; -

W = 340 /;

V = 7 /;

= 21.45. , -

.

53. = 10600 ;

= 600 ; = 734 . .; -

W = 640 /;

V = 13 /;

= 07.28. , -

.

54. = 9100 ;

= 500 ; = 758 . .; -

W = 570 /;

V = 9 /;

= 12.35. , -

.

55. = 7200 ;

= 800 ; = 772 . .; -

W = 340 /;

V = 5 /;

= 14.43. , -

.

() -

. -

, (. 12).

()

. -

.

-

()

-

.

S 1 (t 1) S 2 (t 2)

1 2

1 2 1

. 12.

.

13. = 7800 ; -

= 500 ; = 738 .

.; W = 460 /; -

V = 8 /;

= 10.42. ,

.

. 1. :

. = (760 738)11 = 242 .

2. :

= . = 7800 500 242 = 7058 .

3. (. 13):

t 15 .

S (t)

= 760 .. .

.

t V 1 1 2 2 10 V 10 t

. 13.

4. :

. = t = 10.42 0.15 = 10.27.

5. , ,

:

S = 115 .

14. = 9600 ; -

= 600 ; = 769 .

.; W = 760

/; V = 12 /;

3000 : -

: 400 / 7 /;

= 19.25. , -

.

. 1. ,

:

.1 = 1 = 9600 3000 = 6600 ,

1 = 3000 .

2. :

t 1 9 .

3. -

:

S 1 = 106 .

4. :

. = (760 769)11 = 99 .

5. :

.2 = 1 . = 3000 600 ( 99) = 2499 .

6. :

t 2 6 .

7. -

:

S 2 = 40 .

8. , :

S = S 1 + S 2 = 106 + 40 = 146 .

9. :

. = t 1 t 2 = 19.25 - 0.09 0.06 = 19.10.

:

, . -

-

.

15.

= 10100 . = 400 ,

= 716 . . -

= 11.36. ,

1 = 3000 V = 15 /, W = 710 /.

1 = 3000 6 / 350 /. -

:

30 1500 , 900. -

.

. 1. ,

:

.1 = 10100 3000 = 7100 .

2. :

t 1 8 .

3. -

:

S 1 = 93 .

4. :

. = (760 716)11 = 484 .

5. :

.2 = 1 . = 3000 400 484 = 2116 .

6. :

t 2 6 .

7. -

:

S 2 = 34 .

8. , :

S = S 1 + S 2 = 93 + 34 = 127 .

9. :

. = t 1 t 2 = 11.36 - 0.08 0.06 = 11.22.

10. ,

:

.2 = 1 = 3000 1500 = 1500 .

,

760 . .

, -

.

11. (

):

t 2 4 .

12.

:

S 2 = 24 .

13. -

:

S = S S S 2 = 127 30 24 = 73 .

14. :

t = S/W1 = 370 .

15. :

V.. = .1/t = 19 /.

56. = 6600 . -

= 900 ,

= 735 . . = 16.25. -

,

V = 10 /, W = 450 /.

57. = 5400 . -

= 500 ,

= 743 . . = 18.05. -

,

V = 9 /, W = 410 /.

58. = 6600 . -

= 500 ,

= 722 . . = 19.45. -

,

V = 8 /, W = 450 /.

59. = 8100 . -

= 900 ,

= 723 . . = 16.45. -

,

V = 10 /, W = 610 /.

60. = 5400 . -

= 600 ,

= 773 . . = 18.05. -

,

V = 9 /, W = 410 /.

61. = 9600 . -

= 500 ,

= 735 . . = 16.25. -

,

1 = 3000 V = 14 /,

W = 750 /. 1 = 3000

6 / 380 /.

62. = 9600 . -

= 900 ,

= 755 . . = 11. 45. -

,

1 = 3000 V = 16 /,

W = 800 /. 1 = 3000

8 / 480 /.

63. = 9600 . -

= 600 ,

= 738 . . = 11.22. -

,

1 = 3000 V = 12 /,

W = 680 /. 1 = 3000

6 / 420 /.

64. = 10100 . -

= 500 ,

= 741 . . = 18.57. -

,

1 = 3000 V = 14 /,

W = 840 /. 1 = 3000

6 / 390 /.

65. = 8100 . -

= 500 ,

= 728 . . = 18.25. -

,

1 = 3000 V = 12 /,

W = 750 /. 1 = 3000

8 / 420 /.

66. = 8600 . -

= 700 ,

= 755 . . = 19.05. -

,

1 = 3000 V = 12 /,

W = 720 /. 1 = 3000

7 / 360 /.

: 40 1500 , 1200 . -

.

67. = 10100 . -

= 500 ,

= 726 . . = 21.23. -

,

1 = 3000 V = 15 /,

W = 790 /. 1 = 3000

7 / 390 /.

: 30 1200 , 1500 . -

.

68. = 9100 . -

= 400 ,

= 716 . . = 11.36. -

,

1 = 3000 V = 12 /,

W = 710 /. 1 = 3000

6 / 350 /.

: 40 1500 , 900 . -

.

69. = 8600 . -

= 600 ,

= 732 . . = 10.06. -

,

1 = 3000 V = 15 /,

W = 710 /. 1 = 3000

6 / 330 /.

: 30 1800 , 1200 . -

.

70. = 9100 . -

= 700 ,

= 746 . . = 15.32. -

,

1 = 3000 V = 14 /,

W = 750 /. 1 = 3000

6 / 370 /.

: 40 1500 , 900 . -

.

71. = 8100 . -

= 400 ,

= 716 . . = 08.36. -

,

1 = 3000 V = 14 /,

W = 730 /. 1 = 3000

8 / 410 /.

: 50 4500 . -

.

72. = 9600 . -

= 900 ,

= 715 . . = 18.56. -

,

1 = 3000 V = 16 /,

W = 720 /. 1 = 3000

7 / 400 /.

: 40 4200 . -

.

2.3.

,

-

. ,

.

, , -

, .

, .

-

, -

, .

:

() ().

, -

.

V ,

. -

,

.

V -

-

.

.

V - -

,

. : -

- .

-

= 1,225 /3, = 760 . . t0 = + 15 . -

. -

, .

(. 2), -

- -

. . -

, 9000 500 /,

800 / (60% 500 300).

2

, 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

V, % 5 10 15 20 25 30 40 50 60 70

-

, -

. -

, 350400 / 3000

. , ,

.

(. 3).

3

, /

V = V + V + V + V,

V ;

V ;

V ;

V .

, -

11, 12, 14 15 (. 14). t V 11 14 12 15 760 V.

. 14.

, 12 ,

11 -

V, / , 300 400 500 600 700 800

2000 4000 6000 8000 10000 12000 14000

1 2 3 4 6 9

12

2 4 6 9

13 19 25

3 6

11 17 24 34 48

4 10 18 28 40 56 73

7 16 27 41 56 78 97

9 23 39 53 80 98

118

15 14

.

, .

16. 210 /.

1200 . tH = + 8 .

-

V + V = 5 /. .

. 1. :

V. = V + V = 210 + ( 5) = 205 /.

2. ,

:

V = 217 /.

,

.

17. 160 /.

900 .

tH = + 10 . V + V = 5 /. -

.

. 1. :

V. = 160 + ( 5) = 155 /.

2. ,

:

V = V. ,

%, .

= 900 4,5 % = 0,045.

V = 0,045 155 = 7 /.

3. :

V = V. + V = 155 + 7 = 162 /.

:

V = V V V V.

-

, .

18. 240 /. -

2400 . tH = 15.

V + V = 5 /. -

.

. 1. -

. 11 -

12 . -

14 15 -

.

V. = 220 /.

2. :

V = V. V V = 220 ( 5) = 225 /.

.

19. 200 /.

3000 . tH = 15 . -

V + V = + 8 /.

.

. 1. -

.

,

V = V. + V = V. + V. = V. (1 + ),

= 0,15 = 3000 .

V. = V/(1 + ) = 200/(1 + 0,15) = 200/1,15 = 174 /.

2. :

V = 174 8 = 166 /.

-

, .

, -

, :

V = V + V + V + V + V,

V .

. 3 (. 15).

-154 , -

.

:

V = V V V V V.

,

.

20. = 9600 -

V = 850 /. -

-15 t = 32 . V

= + 8 /. .

. 1.

- . -

.

(. 4)

.

. 15.

4

, t,

V, / 100 200 300 400 500 600 700 800 900 1000 1100

-15 0 1 2 3 4 8 13 19 25 32 41

0 1 2 4 7 10 13 17 21 26 32

:

t = 22 .

2. :

t = 32 22 = 54 .

3. , -

:

V. = 515 /.

4. :

V = 23 /.

5. , -

:

V = 515 ( 23) 8 = 530 /.

:

V = V. + V + V + V.

,

, -

. -

, .

-

(. 16). t V 11 14 13 15 760 V.

. 16.

21. = 6600 . -

V. = 470 /.

V = + 6 /, V = 8 /.

-15 t = 30 . -

.

. 1. :

t = 4 .

2. :

t = 30 4 = 34 .

3. :

V. = V. + V + V = 470 + (+ 6) + ( 8) = 468 /.

4. , -

:

V = 460 /.

73. = 7800 . -

V = 510 /. -

V = - 8 /, V = - 5 /.

t = - 42 . .

74. = 1200 . -

V = 210 /. -

V = + 5 /, V = - 1/.

t = - 12 . .

75. = 8100 . -

V = 610 /. -

V = - 18 /. t = - 42 . -

.

76. = 4500 . -

V = 340 /. -

V = - 5 /, V = 8 /.

t = - 26 . .

77. = 10600 . -

V = 610 /. -

V = - 23 /. t = - 56 . -

.

78. = 600 .

V = 170 /.

V = - 5 /, V = 2 /. t = 14 .

.

79. = 8100 . -

V = 810 /.

V = - 23 /.

t = - 42 . .

80. = 6600 . -

V = 400 /.

V = 4 /, V = 11 /. -

t = - 28 . .

81. = 9600 . -

V = 910 /.

V = 26 /

t = - 35 . .

82. = 7200 . -

V = 470 /.

V = 14 /, V = - 7 /.

t = - 32 . -

.

83. = 1500 . -

V = 190 /.

V = + 5 /, V = - 2 /. -

t = + 12 . .

84. = 7200 . -

V = 570 /.

V = + 15 /, V = - 8 /.

t = - 30 . .

85. = 9600 . -

V = 810 /.

V = - 21 /.

t = - 34 . .

86. = 5100 . -

V = 470 /.

V = - 5 /, V = - 8 /.

t = - 20 . .

87. = 9100 . -

V = 810 /.

V = - 28 /.

t = - 40 . .

88. = 10100 . -

V = 910 /. V

= -23 /.

t = - 36 . .

3. -

, -

. -

. -

(. 17).

- ,

, , , .

,

, , .

,

, -

.

V

U

. 17.

:

. = 180; . = 180 ; . = 180 + ,

;

.

V ,

U , -

,

W :

W = V + U .

,

-

(. 18).

V

U

W

. 18.

W , V , U , -

: (), () (), (), ()

().

,

. -

-

.

,

.

0 360 0 180. ,

-

. +,

.

:

= + ;

= ;

= = + ;

( )W

sinV

sinU

sin +== ; W = V cos + U cos ;

W2 = V2 + U2 + 2VU cos .

, , :

- ( = 0) W = V + U;

- ( = 180) W = V U;

- ( = 90 270) -

, V

U60max , - W V.

, -

,

.

4- :

- ;

- ;

- ;

- .

.

3.1.

, , -

, -

V, , U -

W,

.

:

1. -

( , -

):

. = 180, . = 180 , . = 180 + . 2. :

= .

3. W . -

(. 19). + 3 4 5 U V W . 19.

V 5 -

3 U 5

3 , ( +

) 3 5 W.

4. :

= .

22. , -

, V = 450 /, = 210, U = 60 /,

= 8, . = 144.

. 1. (

):

. = 180 = 210 180 ( 8) = 38. 2. :

= . . = 38 144 = 106.

3. , :

= 7, W = 430 /.

4. :

= . = 144 ( 7) = 151.

.

:

sinVUsin = .

:

sinsinV

U60 max= . ,

(. 5).

5

, 0 6 12 18 24 30 37 45 54 64 > 75 sin 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

:

W = V cos + U cos .

:

W V + U cos .

cos , :

cos = sin (90 ). 23. ,

, V = 360 /, = 140, U = 60 /,

= 12, . = 18.

. 1. (

):

. = 140 + 180 ( 12) = 332.

2. :

= 332 18 = 314 = 46.

3. :

3606060max= = 10;

sin ( 46) 0,7.

= 10 ( 0,7) = 7.

W 360 + 60 cos ( 46) = 360 + 60 0,7 = 360 + 42 = 402 /.

4. :

= 18 ( 7) = 25.

3.2.

,

, -

.

, -

-

(. 20).

:

1. ,

:

W = W V cos .

cos 1. W W V,

.

2. .

W

W

V U V sin

. 20.

,

tg = V sin / W.

(. 21): 3 4 5 W V

. 21. , -

.

3. , -

(. 22):

U = V sin / sin . 3 4 5 U V

. 22.

4. -

:

- , W < V,

= + ;

- W > V,

= + 180 + . 24. V = 630 /; = 18; = 8; W = 650 /; = 9.

.

. 1.

:

W = 650 630 = 20 /.

2. :

= 77.

3. , :

U = 88 /.

4. -

, W > V:

= + 180 + = + + 180 + = = 18 + ( 9) + ( 8) + 180 + ( 77)= = 104

.

(. 20), -

W = U,

U.

,

:

Vsin

Usin = .

sinVUsin =

VUsin = ,

VU60

.

K60VU == .

= arctg (U/U).

, -

, 40, 5 0,1 (. 6).

6

, 0 5 10 15 20 25 30 35 40 45 tg 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 1,0

, U < U.

U > U, -

:

= 90 arctg (U/U).

:

U = U / sin .

, :

- , W < V,

= + ;

- W > V,

= + 180 + . 25. V = 540 /; = 342; = 5; W = 500 /; = 8.

.

. 1.

:

W = U = 500 540 = 40 /.

2. :

60540

U = ( 5) = 45 /.

3. . U > U, -

:

= [90 arctg (40/45)] = [90 arctg (0,9)] = (90 42) = 48.

4. :

U = 45/ sin 48 = 45/0,7 = 64 /.

5.

= 342 + ( 8) + ( 5) ( 48) = 17.

3.3.

-

, .

,

, , -

,

(. 23):

V + U = W,

U = W V. U

V U

W

V U

W

. 23.

.

-

, .

:

U = U cos (U2/2V)sin2 ,

U = U U2 /2V.

-

(. 7).

-

. -

.

:

i

ni

1ii

. SUS1U =

== ,

Si i- ;

Ui i- ;

n .

7

V = 400500 / , / -

, 20 40 60 80 100 120 140 160 180 200 0, 360 5, 355 10, 350 15, 345 20, 340 25, 335 30, 330 35, 325 40, 320 45, 315 50, 310 55, 305 60, 300 65, 295 70, 290 75, 285 80, 280 85, 275 90, 270

95, 265 100, 260 105, 255 110, 250 115, 245 120, 240 125, 235 130, 230 135, 225 140, 220 145, 215 150, 210 155, 205 160, 200 165, 195 170, 190 175, 185 180, 180

20 20 20 19 19 18 17 16 15 14 13 11 10 8 6 5 3 1 -1 -2 -4 -6 -7 -9 -10 -12 -13 -14 -15 -16 -17 -18 -19 -19 -20 -20 -20

40 40 39 38 37 36 34 32 30 27 25 22 19 15 12 9 5 2 -2 -5 -9 -12 -15 -18 -21 -24 -27 -29 -31 -33 -35 -36 -38 -39 -39 -40 -40

60 60 59 58 56 54 51 48 44 40 36 32 27 22 17 12 6 1 -4 -9 -14 -19 -24 -29 -33 -37 -41 -44 -47 -50 -53 -55 -57 -58 -59 -60 -60

80 80 78 77 74 71 68 63 58 53 47 41 35 28 21 14 7 0 -7 -14 -21 -27 -34 -40 -45 -51 -56 -60 -64 -68 -71 -74 -76 -78 -79 -80 -80

100 100 98 96 93 89 84 78 72 65 58 50 42 33 24 16 7 -2 -11 -20 -28 -36 -44 -51 -58 -65 -71 -76 -81 -86 -89 -93 -95 -97 -99 -100 -100

120 120 117 115 111 106 100 93 85 77 68 58 48 35 27 16 5 -5 -16 -26 -36 -46 -55 -66 -72 -80 -87 -93 -99 -104 -108 -112 -115 -117 -118 -120 -120

140 140 137 134 129 123 116 107 98 88 77 66 54 41 29 16 3 -9 -22 -34 -45 -57 -67 -77 -86 -95 -103 -110 -116 -122 -127 -131 -134 -137 -138 -140 -140

160 160 156 152 147 140 131 122 111 99 86 73 59 44 30 15 0

-14 -28 -42 -55 -68 -80 -91 -101 -111 -120 -127 -134 -140 -146 -150 -153 -157 -158 -160 -160

180 180 176 171 165 157 147 135 123 109 95 79 63 46 30 13 -4 -20 -36 -51 -66 -80 -93 -106 -117 -127 -137 -145 -153 -159 -165 -169 -173 -176 -178 -180 -180

200 200 196 190 183 173 162 149 135 119 102 85 67 48 29 10 -8 -27 -44 -62 -78 -93 -108 -121 -133 -145 -155 -164 -171 -179 -184 -189 -193 -196 -198 -200 -200

89. , , -

. / , V, / , U, / , S,

1 25 180 330 30 5 36

2 48 370 210 50 7 74

3 74 420 160 70 12 98

4 118 450 270 80 8 57

5 176 540 40 60 9 69

6 235 630 260 90 16 125

7 270 710 130 70 5 238

8 303 780 80 120 4 146

9 349 840 310 100 10 347

10 04 920 90 90 5 642

90. -

. / , V, / , W, / ,

1 124 770 - 8 740 5

2 259 360 6 390 13

3 43 180 4 160 10

4 344 860 - 5 790 4

5 16 940 12 850 5

6 96 410 - 4 500 13

7 235 560 12 520 14

8 282 900 - 7 860 14

9 216 510 9 610 15

10 324 850 -10 930 9

91.

.

/ , V, / , U, / ,

1 58 490 200 100 12

2 276 480 130 60 10

3 87 520 70 100 15

4 148 400 190 60 9

5 215 410 30 120 7

6 325 480 240 40 6

4.

-

,

.

.

-

.

-

, -

.

, , - -

.

-

,

: -

, .

-

, , -

(. 24).

,

.

, -

, , , -

.

,

, ,

, .

, -

-

.

, .

, ,

, -

, , -

. 1

. 24.

:

- ;

- , ;

- ,

;

- .

.

4.1.

,

. -

.

(.25).

= + 180 = + ( = 180).

= , , < ,

- , > , - .

3

2

1

-

+

. 25.

-

:

= .

, = :

=

= 180.

26.

= 245, = 238. = 190.

, .

. 1. :

= 238 + 190 180 = 248.

2. :

= 248 - 245 = 3.

3. :

= 190 180 = 10.

, -

:

= S tg .

,

: .

b ( 2b), -

, ,

, 2090 (.26).

:

= .

:

, +.

:

= 180 + .

. 26. , :

=

= .

:

= 180 + ; = .

27.

= 24, = 28. 8

510 / = 170.

2b = 10 . , , = 40,

.

. 1. :

= 28 + 170 180 = 18.

2. :

= 18 24 = 6.

3. :

= 170 180 = 10.

4. -

(. 27):

S = 68 .

S W 1 2 t

. 27.

5. :

= 7 .

6. :

= 24 ( 40) = 64.

7. :

= 180 40 = 140.

8. :

= 24 ( 10) = 34.

9. :

= 180 + ( 10) = 170.

,

, .

. -

:

= (S/S) = (t/t) .

:

= + .

-

(. 28): 4 5 S (t) S (t) S (t)

. 28.

(. 29):

= .

S S

. 29. 28. = 164

= 168. 12 -

= 188.

17 . .

. 1. :

= 164 + 188 180 = 172.

2. :

= 172 168 = 4.

3. :

t = t t = 17 12 = 5 .

4. :

= 9, = 13.

5. :

= 164 13 = 151.

92. = = 128. 10

= 192. , = 40, .

93. = 32. = 26. 7 = 188. , = 20, .

94. = 229. = 225. 12 = 178. , - = 40, .

95. = 86. = 88. 15 = 168. , = 50, .

96. = 148, = 138. 9 = 182. , = 40, .

97. = = 294. 15 = 176. 6 . .

98. = 224. = 232. 10 = 180. 8 . .

99. = 336. = 344. 15 = 186. 8 . .

100. = 348. = 340. 10 = 172. 7 . .

101. = 22. = 26. 9 = 168. 7 . .

102. = 168. . = 0 54 . 19.27 = 172. 19.53 - = 182. , .

103. = 185. . = 1 05 . 06.32 = 192. 06.50 - = 178. , .

104. = 292. . = 0 31 . 14.23 = 296. 14.43 - = 174. , .

105. = 86 . , 55. = 185 , = 40. - .

106. = 12 . , 42. = 175 , = 60. - .

107. = 126. = 70. 10 = 175 = 160. .

108. = 352. - = 310. 9 = 188 = 30. .

109. = 84. = 110. 12 = 188 - = 50. .

110. = 52. - = 75. 11 = 194 = 30. .

4.2.

-

(. 30).

>

=

<

S S

. 30.

,

= + = + ( = 360).

= , > -

, < - .

, .

:

= .

-

:

= (S/S) = (t/t) .

, .27.

:

= + .

-

, ,

.

, -

, (. 31).

S S

. 31. (2090) -

:

= .

.

= :

= 360 + .

,

:

= ;

= 360 + .

29.

= 247, = 254. 12

400 / = 348.

14 . 2b = 10 . -

, , = 60,

.

. 1. :

= 254 + 348 360 = 242.

2. :

= 247 242 = + 5.

3. , :

= + 6.

4. :

= 247 60 = 187.

5. :

= 360 + 60 360 = 60.

6. :

= = 247 254 = 7.

7.

= ( 7) + (+ 6) = 1.

8. :

= 247 ( 1) = 248.

9. :

= 360 + ( 1) = 359.

, -

(.32).

S S

. 32.

, .

:

= .

:

= 360 + .

30. = 338

= 333. 7

84 = 348.

116 . .

. 1. :

= 338 + 348 360 = 326.

2. :

= 333 326 = 7.

3. :

S = S S = 116 84 = 32 .

4. :

= 2, = 9.

5. :

= 338 9 = 329.

111. = 274, = 278. 11 = 8. 24 . , = 60, .

112. = 232. = 234. 9 68 = 354. - 174 . ,

= 30, . 113. = 85. = 88.

10 = 10. 118 , 330 /.

, = 70, . 114. = 335. = 328.

9 = 6. , = 30, . 19 .

115. = 18, = 25. 10 = 12. 84 , 205 . , = 50, .

116. = 22. = 26. 9 56 = 352. - 146 . .

117. = 224. = 222. 10 = 6. 6 -. .

118. = 82. = 92. 15 = 354. 9 . .

119. = 55. = 60. 12 88 = 12. 137 . .

120. = 194. = 192. 10 = 356. 4 . .

4.3.

:

- ;

- .

:

= + 180 + , .

:

= ( ) sin ,

, = ( + )/2;

, ;

, .

, -

100150 , -

, -

.

,

, .

, 90, , , -

30 150.

.

,

, (. 33).

. -

.

1 . C

C

1 (1)

C C

2 (2)

1

2

. 33.

, 2 1 > 1 , -

-

(. 34):

S = W (2 1).

2 (10.14)

1 (10.11)

S

(10.14)

. 34.

.

100150 , 300 . -

, 2530. -

,

.

, , -

. , -

.

(. 35).

,

. .

W, .

1 2 15.25 (15.25)

15.12

S

. 35.

6. -

:

-

;

-

;

- (, , -

);

- -

, -

.

-

,

,

.

, - ,

-

.

:

-

;

- ;

- ;

- ( , -

);

- , ;

- ;

- -

;

-

.

-

, . -

250 350 , 150 200 , -

100200 , 100150 ,

150 200 .

-

, ,

(. 36):

- ,

;

- - , -

, , -

= + = + ( ) + .

, -

, -

, ,

= + 180 = + ( ) + 180.

150200

,

;

- -

;

-

= 22 HH . , > 5, , = .

. 36. , -

. 150250

. -

, , -

.

.

,

(. 37).

15 . :

= ,

,

.

+ ,

, , - .

0 360, , -

:

= 360.

S S. S'

S. () 1 S 2 1

S S 2

. 37.

= arcsin (S./S)

(. 38). 3 4 5 S S. S

. 38.

(. 8).

8

(S. = 15 )

S, 50 60 70 80 90 100 110 120 130 140 150

, 18 15 13 11 10 9 8 7 7 6 6

:

= + .

, -

50 .

:

S' = S sin(1 + 2) = S sin,

S .

-

, ,

= 500 (. 39).

D

R R

. 39. ,

.

:

= + ,

;

-

, .

-

:

= + .

:

= 17,5 (') + 0,062 + 500,

, ;

' , ' = A 0,5b;

A

;

b -

b 3.

:

.t

V;V

t

.

=

=

-

, -

, .

31. = 236. 120

, -

1 = 4 2 = 35.

.

. 1. -

, (. 38):

S = 8,4 .

2. :

= 7.

3. :

= = 4 7 = 3.

4. :

= 236 3 = 233.

32. , -

= 162. 100

, -

1 = 346, 120

2 = 22. -

.

. 1. :

= 14 + 22 = 36.

2. :

S' = 100sin 36 = 57 .

3. :

S = 24 .

.

33. , -

= 28. 80 -

, 1 = 352

2 = 332, 90 3 = 25

4 = 54. -

.

. 1. :

= 8 + 25 = 33.

2. :

S' = 80sin 33 = 44 .

, S' < 50 .

, .

3. :

1 = 332 360 = - 28;

2 = 54.

4. . 8

= 11.

5. :

= 28 + 11 = 39.

6. :

= 28 - 39 + 360 = 349.

121. 08.15, = 44, = 28, S = 80 ,

= + 11, = 85, = 84, = 65. 122. 21.14, = 184, = 332, S = 130

, = + 6, = 153, = 151, = 58. 123. 20. 36, = 348, = 325, = + 8,

S = 60 , = 77, = 76, = 72. 124. 06.43, = 16, = - 5, = 45,

S = 150 , = 34, = 36, = 60. 125. 11.55, = 32, = 300, S = 100 ,

= - 7, = 84, = 83, = 52. 126. 19.28, = 242, = 334, S = 60 ,

= + 9, = 54, = 53, = 78. 127. = 308. 105

1 = 6 2 = 345. .

128. = 86. 80 1 = 348 2 = 15. - .

129. = 122. 110 1 =356 2 = 315. 130 : 3 = 6 4 = 28. - .

130. = 16. 90 1 =334 2 = 345. 120 : 3 = 356 4 = 28. - .

131. = 160. 100 1 =314 2 = 346. 110 : 3 = 18 4 = 42. - .

132. = 334. 120 1 = 318 2 = 342. 110 : 3 = 356 4 = 36. - .

6.

34. 9100

580 / tH = - 52 C,

= 124. V = + 12 /.

117. 8 115 .

356. 254 .

+ 12.

( ).

. 1. -

:

V = 33 /;

V. = V + ( V) + ( V) = 580 + 12 33 = 559 /. 2. (. 11), -

:

V = 890 /.

3. , -

. 28:

W = 860 /.

4. :

W = W V = 860 890 = 30 /.

5. :

= = 117 124 = 7.

6. :

= + = 124 + 356 360 = 120.

7. :

= = 117 120 = 3.

8. :

S = S S = 254 115 = 139 .

9.

(. 29):

4.

10. :

= + = ( 7) + ( 4) = 11.

11. (. 18):

= 80.

12. (. 19):

U = 163 /.

13. ,

:

= + + = 124 + ( 11) ( 80) + (+ 12) = 205.

35. 10.24 = 168, = 172. 12 = 12, - 92 . 10.52.

( ), V = 440 /, = + 6. . 1. , :

W = 460 /.

2. :

W = 460 440 = 20 /.

3. :

= 168 + 12 = 180. 4. :

= 172 180 = 8. 5. :

t = 10.52 10.24 0.12 = 0.16.

6. :

= 8 16/12 = 11. 7. :

= 172 168 = 4. 8. :

= 4 + ( 11) = 7. 9. :

= 70.

10. :

U = 57 /.

11. :

= 168 + ( 7) + 180 + ( 70) + 6 = 277.

36. 08.25. V = 440 /. -

08.45. S = 156 . = 08.58.

S = 110 . -

.

. 1. -

:

t = 08.45 08.25 = 0.20.

2. :

W = 468 /.

3. :

t = 08.58 08.45 = 0.13.

4. :

W = 508 /.

5. -

:

W = 508 468 = 40 /.

6. :

V = 440 + 40 = 480 /.

37. = 4800 .

V = 410 /. -

V = 8 /, V = 12 /.

t = 18 . 12

82 . 218 , -

33 . ,

.

. 1. (. 15)

:

V = 4 /.

2. :

V. = 410 + 12 8 4 = 410 /.

3. 10 = 4800 t = 18

:

V = 520 /.

4. :

W = 410 /.

5. :

S = 218 82 = 136 .

6. :

t = 0.33 0.12 = 0.21.

7. :

W = 390 /.

8. -

:

W = 390 410 = 20 /.

9. :

V = 520 20 = 500 /.

10. :

V. = 395 /.

11. :

V. = 395 ( 4) ( 8) 12 = 395 /.

133. 8100

520 / tH = 48 C,

= 236. V = 22 /.

243. 12 155 . -

186. + 8. -

( ).

134. = 338, = 342. 18.47 . 15 6 , -

105 . (

), V = 470 /, = + 15. 135. 10600

540 / tH = 58 C,

= 336. V = + 14 /.

333. 7 102 , -

284 .

352. + 12. (-

).

136. 10.48. V = 840 /.

10.56. S = 112 . = 11.10.

S = 187 . -

.

1. = 41. 2. = 20; = 12. 3. = 166; = 176; = + 10.

4. = 332; = 334; = + 2. 5. = 84. 6. = 12; = 3.

7. = 280. 8. = 140; = 123; = - 17. 9. = 47; = 37.

10. = 166. 11. = 285; = 281. 12. = 324; = 326; = + 7.

13. = 15; = 19; = + 7. 14. = 89; = 84.

15. = 93; = 103; = + 10. 16. = 75. 17. = 0.

18. = 10; = - 4. 19. = 251; = + 3. 20. = 354.

21. = - 2. 22. = + 6. 23. = + 4. 24. = + 7. 25. = - 3.

26. = 1348 . 27. = 1212 . 28. = 968 . 29. = 7904 .

30. = 7230 . 31. = 10400. 32. . = 338 . 33. = 620 .

34. = 300 . 35. = 400 . 36. = 600 . 37. = 600 .

38. . = 380 . 39. . = 792 . 40. . = 558 .

41. . = 2958 . 42. . = 1031 . 43. . = 1220 .

44. . 760 = 905 . 45. . 760 = 1494 . 46. . 760 = 3920 .

47. . 760 = 1003 . . = 1500 . 48. . 760 = 1333 ; . = 1800 .

49. . = 2700 . 50. . = 4200 . 51. = 10.51.04; S = 130 .

52. = 21.51.22; S = 47 . 53. = 07.40.26; S = 133 .

54. = 12.50.53; S = 151 . 55. = 15.04.46; S = 123 .

56. . = 16.34.03; S = 68 . 57. . = 18.13.44; S = 60 .

58. . = 19.56.50; S = 89 . 59. . = 16.56.19; S = 115 .

60. . = 18.14.09; S = 63 . 61. . = 16.10.58; S = 137 .

62. . = 11.43.52; S = 126 . 63. . = 11.06.50; S = 146 .

64. . = 18.42.11; S = 160 . 65. . = 18.13.26; S = 120 .

66. . = 18.51.52; S = 125 ; V = 17,5 /.

67. . = 21.12.05; S = 137 ; V = 18,3 /.

68. . = 11.21.39; S = 134 ; V = 17,3 /;

69. . = 10.53.58; S = 106 ; V = 19,2 /.

70. . = 15.18.46; S = 128 ; V = 17 /.

71. . = 08.25.31; S = 104 ; V = 13,5 /.

72. . = 18.45.18; S = 109 ; V = 15,7 /. 73. V = 706 /.

74. V = 219 /. 75. V = 855 /. 76. V = 418 /. 77. V = 945 /.

78. V = 170 /. 79. V = 600 /. 80. V = 275 /. 81. V = 555 /.

82. V = 325 /. 83. V = 170 /. 84. V = 568 /. 85. V = 780 /.

86. V = 450 /. 87. V = 760 /. 88. V = 875 /.

89.

/ 1 2 3 4 5 6 7 8 9 10

, - 7 3 - 9 - 4 4 - 5 3 - 5 3 - 6

W, / 160 410 420 520 590 570 770 870 750 900

t, 14 11 14 7 7 13 24 10 28 43

, 32 45 85 122 172 240 267 308 346 10

90.

/ 1 2 3 4 5 6 7 8 9 10

, 185 149 5 22 327 242 190 331 69 79

U, / 113 48 24 104 220 95 125 117 129 182

91.

/ 1 2 3 4 5 6

U, / 88 54 - 100 - 40 117 - 9

92. = 88; = 220; = 116; = 192.

93. = 12; = 200; = 24; = 188.

94. = 269; = 140; = 227; = 178.

95. = 136; = 130; = 98; = 168.

96. = 98; = 220; = 136; = 182. 97. = 308.

98. = 206. 99. = 342. 100. = 348. 101. = 44.

102. = + 6; = 160. 103. = + 5; = 185.

104. = - 2; = 302. 105. = 15; = 220.

106. = 102; = 120. 107. = 146. 108. = 142.

109. = 214. 110. = 208.

111. = 338; = 300; = 283; = 355.

112. = 202; = 30; = 228; = 4.

113. = 155; = 290; = 94; = 351.

114. = 305; = 30; = 327; = 8.

115. = 75; = 310; = 24; = 1.

116. = 115. 117. = 228. 118. = 98. 119. = 66.

120. = 184. 121. = 264. 122. = 344. 123. = 142.

124. = 234. 125. = 146. 126. = 46. 127. = 322.

128. = 63. 129. = 157. 130. = 41. 131. = 160.

132. = 18. 133. = 121; U = 96 /. 134. = 311; U = 76 /.

135. = 291; U = 108 /. 136. V = 800 /.

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