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2007 41 & , .: 210 3608065, Fax: 210 3608197 www.volonaki.gr, mail: [email protected] : : :
, , . ISBN 978-960-381-362-0
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9
13
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, 1,2 , 0,5 . : 5 , 1,2 + 0,5 5 = 3,7 8 , 1,2 + 0,5 8 = 5,2 10 , 1,2 + 0,5 10 = 6,2 , 1,2 0,5 ( ). x : = 1,2 + 0,5x x , . O , , . . . (, , , x, y, z, t, ...) , . 5 (2) + 8 : 2 3 .
17
7x + 3 + 2y + = ( + ) = ( ) . . ) 3 x + 5 x = (3 + 5) x = 8 x ) 5 + 7 = (5 1 + 7) = 11 ) y + 13 y + 5 = (1 + 13) y + 5 = 14 y + 5 , (.) . () . 7x 3y 7 x 3 y 51 (3x 2y) 51 (3 x 2 y)
1) ) 5 3 + 18 ) x + 3x 7x + 10x ) 4 + 5x + 2x ) y 3 8y + 6 2
t t t t t
18
: ) 5 3 + 18 = (5 3 + 18) = 20 ) x + 3x 7x 10x = (1 + 3 7 + 10) x = 7x ) 4 + 5x + 2x = (4 1) + (5 + 2) x = 3 + 7x ) y 3 8y + 6 2 = (1 8) y + (3 + 6) 2 = 7y + 3 2
2) = 2 (x 2y + 1) 3(2x y 3) 4x x = 1 y = 1 : = 2 (x 2y + 1) 3 (2x y 3) 4x = 2x 4y + 2 6x + 3y + 94x = (2 6 4)x + (4 + 3)y + 11 = 8x y +11 O x = 1 y = 1 : = 8 (1) 1 + 11 = 8 1 + 11 = 18 3) = 17 [5 + 3 + ( + 9)] + = 12 = 17 (5 + 3 + ( + 9) = = 17 (5 + 3 + 9) = = 17 + 5 + 3 + 9 = = ( + ) = ( 12) = 12 4) ,
1
= 2x + x + y x y 2y : = 2x + x + y x y 2y = 2x 2y + x y + y x = 2 (x y) + (x y) + (y x) = 2 (x y) + (x y) (x y) = = = =11 =0
19
t t t t t
1) , . . 7x 4x + x 1. 5x . x + 3x 5x 2. 4x . 3 2x + 7x 3. 8x . 3x + 4x 9x 4. x 5. 7x 6. 2x 2) () () ) 15 4 + = 11 ) 2(x 1) + (x + 2) = + x ) (3 + 7) (2 + 6) = + 1 ) y + (y 2) + (3y + 2) = 3y + 2 ) 7x (x + 1) (6x + 6) = 0
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) 13x 7 + 3y x 2y + 7 (11x + y)
3. . ) = 5x 2(6 3x) + 4(2 + x) x = 3 ) = 7 ( 2) 2(3 + 3) + 5 = 7 = 1 ) = 19 2( ) (2 x) ( + 12) x = 5 = 6 4. : ) = 5( 2) 3(2 3) + 7 + = 8 ) = x + 2(3x y) 4(y 3) + x ) = 3 + (x) (y ) ( 1), x + y = 11 5. : = ( ) (2 + ) ( +) + ( 2) .
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u
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x. .. 3 x + 4 = 5, 8(x 2) + 3 = 5x 7, 7x 1 = 4x + 6 = =
21
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1) : 7x + 8 = 4 3x : 7x + 8 = 4 3x 7x + 3x = 4 8 4x = 4
t t t t t
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x = 1 2) : 7 (x + 5) = 2(x 1) 3
23
: 7 (x + 5) = 2(x 1) 3 7x + 35 = 2x 2 3 ( ) 7x 2x = 2 3 35 ( ) 5x = 40 ( ) ( ) x = 8 3) : : =3 ( , = 3) (y + 1) ( 2y + 1) = 21 y + 1 2y 1 = 21 ( ) y 2y = 21 1 + 1 ( ) y = 21 ( ) ( ) y = 21 4) : :
24
3 (3x 1) 2 (2x 5) = 1 (5x + 1) 9x 3 4x + 10 = 5x + 5 9x 4x 5x = 5 + 3 10 0x = 2
5) : :
1
10 3 5 ( + 1) = 5 ( 5 ) 30 5 5 = 25 5 5 + 5 = 25 30 + 5 0 = 0 , . 6) (x 2) + 5 = 2 (x ) . : (x 2) + 5 = 2 (x ) x 2 + 5 = 2 x 2 x 2x = 2 + 2 + 5 ( 2) x = 5 2 x , 2 = 0 = 2. 7) , 4x 3 = x . : 4x 3 = x 4x x = 3 (4 )x = 3 4 = 0 3 = 0 = 4 = 3 8) : (x 2) (3x 5) (7 + 4x) = 0
25
, . : = 0 = 0 = 0 . : (x 2) (3x 5) (7 + 4x) = 0 x 2 = 0 3x 5 = 0 7 + 4x = 0 x=2 3x = 5 4x = 7
9) x + 5 = 3x 7 ) = 5, x = 6 ) x = 3, = 1 ) = +3, ) = 5 5x + 5 = 3x 7 5x 3x = 75 2x = 12
x = 6 ) x = 3, 3+5=337 3 = 9 7 5 3 = 3
= 1 ) = 3 3x + 5 = 3x 7 3x 3x = 7 5 0 x = 12 .
26
t t t t t
1. : ) 13 + = 25 ) 7 = 42 ) 3 + 2 = 23 ) 256 = 187 ) 47 = 52 ) 9 + = 4 2. () (). ) 5x + 15 = 0 3 ) 2(x + 1) = 2x + 2 ) 3 (2 x) = 5 3x ) 5x 7 = 2(2x + 3) + x ) O 7 + x = 2 2 x = 7 ( ) ) x = 6 + 4x = 4 3. . ) 3 x = 6 i) 4 ) 7 x = 28 ii) 5 ) ) 4x 5 = 3x iii) 3 iv) 2 v) 10
1
1. : ) 5x 7 = 8 + 2x 3 ) 2(x 3) + 9 = 5x 6 ) 9x 3(2x 5) = 21 ) 8(x 4) 6 (2 x) = 2(6x 1) 2. : ) 4(3 x) 2(3x 4) = (16 x) ) 2(3 3y) 3(1 y) = y 1
t t t t t
27
) 3 2(3x + 1) = x 5 (5 7x) ) 6( 1) (3 + 11) = 7 3. : ) x {3 + [x (x + 3)]} = 5 ) x [ (3x + 1) 5] = 2(x + 1) ) {2(x 4) 3(x + 1) + [10 2(x + 1) 60]} = 15(x + 1) 4. : )
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29
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11. : ) (5x 7) (3x + 12) (x 2) = 0 ) (x + 3) (2x 13) (7x + 3) = 0 ) (x 2) (x2 + 5) = 0 12. (1 x) + 3 = 2x + 5 + ) = 5, ) x = 3, ) = 2,
30
13. , 7x + 5 = x + : 7x + 5 = x + :
) ) 14. . ) ( 2)x = 9 ) 5x = 3 x ) 2x + 7 = 2x + 6 ) 15. x, . ;B A
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3x 20
3x 10
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A 16. x, y, z. 5x 2
3y 5
10cm
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31
) , 75,36cm2 4cm. ) , 125,6cm2 = 5cm. = 2 , 2 , , , 2 ,
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= 75,36 = 4
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= 125,6 = 5
2) , , . : ) 35cm2, 8cm 6cm. ) 80cm2, 12cm 8cm. 2 = ( + ) (1) , + ,
32
(1) 2=+ 2 = ) = 35, = 8 = 6cm
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= 80, = 12 = 8
1. .
t t t t t
2. = h + : i) h ii) 3. = + t i) t ii) 4. Q = m c , m. 5. .
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33
9. i) m ii) r 10. V = V0(1 + a ) . 11. F = 1,8C + 32, F C . C C F = 73,4. 12. = 2(x + y), x y . x y. : ) x 56cm y = 10cm ) T y 148cm x = 46cm.
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34
t t t t t
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x = 15 15 3 15 = 45 . 3) - 4 .
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35
:
180 . 4) O , 360 . , , . O x . 2x 3x. O 2x + x + 3x = 360 6x = 360
36
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3000 . 5000 , . x .
37
40000 6000 = 16000 + 6000 = 22.000
40000 +
40000 + 3000 = 10000 + 3000 = 13.000 .
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t t t t t
1) 2 47. ; . 5x + 47 = 2 . 2x + 5 = 47 . 5x + 2 = 47 . 5x 2 = 47 2) 13 x . ; ; . 2(x + 1) = x 13 . x + 13 = 2(x + 1) . 2x + 13 = x + 2 . x + 13 = 2(x 1)
t t t t t
38
1. , , 5, 9.
2. 69. . 3. , 91. -
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5. 125 17 6. 6. 20 6 , , 3 . O 75 . ; 7. , . 8. 400 . . ; 9. 27 . . 10. 50cm2 , .B E
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11. 120cm. 10cm 10cm -
39
100cm2. . 12. O 50 . 8 . . 13. 48 18 . . 14. , 5 , 10 . 15. , 13 . 2,5 Km/h 4 Km/h, . 16. 8 .. 80 Km/h. 1 100 Km/h. . ( 512 Km). 17. . O , 30
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. 18. O 7 cm. 66 cm, . 19. . 10 cm 120 cm2, cm .
40
20. 48 . 114 . 21. O 12 30 . , 1,5 3 ; 22. . 6 , . . 23. 8 , 6 . , ; 24. 6 , 3 2 . . 25. 12 . 50 , 14 1 . . 26. 5 l 80% 3 l 20% . . 27. ,
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42
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t t t t t
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19 (x + 9) 8(x 1) 19 x 9 8x 8 x 8x 8 19 +9 9x 18( )
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6 (3x 12) 3(5x 1) 10 (x + 5) 18x 72 15x + 3 10x + 50 18x 15x 10x 50 + 72 3 7x 119
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.
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2 (2x 1) (2x 5) > 2 (x + 2) 4x 2 2x + 5 > 2x + 4 4x 2x 2x > 4 + 2 5 0x>1 5) : 2(x 1) 3(x + 1) 2(x + 3) 3(x 3) + 6 < 2(x + 3) : 2(x 1) 3(x + 1) 2(x + 3) 3(x 3) + 6 < 2(x + 3) 2x 2 3x 3 2x 6 3x 9 + 6 < 2x + 6 2x 3x + 2x 6 + 2 + 3 3x 2x < 6 6 + 9 x1 x x + 100 ) 7x 4 > 8x 3 x > 1 3) . . 0x>3 i. . 0 x > 5 ii. x . 0x 5y 13 ) 5 ( 2) 3 + 2 ) 27 (2x + 7) 7(x 1) ) 17x (5x + 3) 4x 3 3. : ) 2(3x 6) < 6(2 + x) ) 4(x + 1) 3(2x 3) 5 ) 4(x 3) 2(3 2x) > 3(x 1) ) 2(3 x) + (x + 5) 4(1 x) ) 8x 3(x 1) 6x 5 ) 5(y + 3) 4(y + 2) < 3(y + 5) 5(y + 2) ) 5 (3 5) 3( 7) 8 2(3 + 4) 4. : ) 3{x 5[ x (x + 2)]} < 15(x 3) ) 11x {(8x + 23) 2(x 8)] 5(3x 7) ) )
48
)
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49
8. : ) 11 3x + 2 < 29 ) 2 < 3 5x < 18 ) 9 8x + 1 13 9. , = 5( 2) 30 ; 10. , 7x 5 + 2 > (x 2) x = 4; 11. 10 . 40 . ; 12. 55 65 13 7. . 13. : 750, 126. ; 14. , 18, 16, 17, 18, 19 20. 15. . : 1 : 100 0,5 . 2 : 150 0,3 . 2 .
50
1 1 . i) ; ii) ; . - . 0 x = 2 . 3(x+2) = 6 . 2 ( 2) x = 0 . ( + 3) x = + 5 = 3 = 5 2 ) :
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51
2 . : . x 13 = 2x + 7 ; 3 . : . 5 + x 1 , 2, : < x
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174
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: B2 = 2 + 2 2 = 12 + 12 2 = 2
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30 30 2cm 2cm
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176
2
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1
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177
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209
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L (,) -
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t t t t t
1) 15,7 cm, . : L=2
217
15,7 = 2 3,14 15,7 = 6,28 =
= 2,5 cm.
2) 30cm. 942 m. L = 2 = 2 3,14 30 = 188,4cm = 1,884m 942 m 942 : 1,884 = 500 3) =6cm B=8cm. . , = 90 6cm 0 8cm B
. : 2 = 2 + 2 2 = 62 + 82 2 = 36 + 64 2 =100 = = 10cm, = 10cm. : L= = 10 3,14 = 31,4 cm.
1. : o o L 6 cm 5,024m 8 cm
t t t t t
113,04m
218
2. () ().
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3
1. 18cm.
t t t t t
2. 81,64 cm, . 3. 26cm. . 4. 8cm. : ) ) . 5. 68cm 3.500 . .A
6. = 12cm = 16cm. .
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219
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t t t t t
1) 100,48cm. 45. L = 2 100,48 = 2 3,14 100,48 = 6,28 45 : = 16 cm.
2) : : 6cm 60 10cm 90o 30o 6,28cm
:
:
221
6cm 60 6,28cm
10cm 90o 15,7cm
12cm 30o 6,28cm
t t t t t
1. . 30 60 90
120 135 150 240
222
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3
1. : 45 /12 75 4/15
t t t t t
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2. 50,24cm : ) ) 90. 3. : L 4cm 30o /4 6cm 5m 270
4. 30cm. 5. 40 8,37m. N .
223
6. 50, 6( 2) + 3 = 4x 1. 7.
= 4cm = 2cm
8. 5cm 8cm. 3. 6.
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1
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t t t t t
1) 94,2cm . . L = 2
224
94,2 = 2 3,14 94,2 =6,28 =
= 15cm
= 2 = 3,14 152 = 3,14 225 = 706,5cm2. 2) To 50,24cm2. . : E = 2 50,24 = 3,14 2 2 = = = 4cm. : L= 2 = 2 3,14 4 = 25,12cm. 3) 50m. 4m. m2 . 1 = 4m 2=50m 2=25m .. = .. = 3,14 (252-42) .. = 3,14 (225 16)=3,14 209=656,26m2 2 = 16
3
25m 4m
1) : L 2cm 25,12cm 153,86cm2
t t t t t
2) . , :
225
: 2 : 4 : 8 : . 3) L, : : : : :
. 4) 4 cm, : : = 2 L : = 4L : E = L : = .
t t t t t
1. 1256 cm2. 2. 81,64m. 3. (,) = 3 cm. (,).
4. .
5cm
3cm
A
5. . (=4 cm = 3 cm)
4cm M 3cm B
226
6. 32cm2. .
7. 16cm. . 8. 615,44cm2 . .
3
9. . R .
R
10. .
8cm
u
3.6 O Y E
ttt0
.
A
B
.. : .. = rad : .. =
227
. , = =
0 A B
t t t t t
1) 94,2 cm, 120. : L = 2 94,2 = 2 3,14 94,2 = 6,28 = = 15cm
: .. = 2) . : =
10cm A 90 10cm B
228
3
= 78,5 50 = 28,5cm2A 12cm
3) . .
3cm E2 6cm
E1
E3
B
12cm
3cm
=12 cm, 1 6 cm
= 6cm, 2 3cm
3
E = 1 + 2 + 3 = 56,52 + 14,13 + 108 = 178,65 cm2.
t t t t t
1) 16cm. 45 : : 8 (cm2) B: 16 (cm2) : 24 (cm2) : 32 (cm2) . 2) 4,71cm2 3cm, : : 90 : 45 : 60 : 120 3) , :
229
: 4
:
:
: 2
4) () () . ) ) ) 2x A 1 0 x 2 B
) 1 = 2 )
1. .
t t t t t
.
2. 120 84,78cm2. . 3. 1519,76cm2. 72. 4. 30 L
.
5. 12 cm = 60. N : ) ) ) .A
60 12cm B
230
6. (,10cm) = 60.
A 60 0
3
7. 1 = 5cm 2 = 12cm. .
0 A 60
8. .(A = 10cm, = 4cm).A 10cm A 4cm
9.
A
10. , 10 cm.
11. ( = 90) = = 6 cm. .
A
231
12. . : ) . ) . ) .
5cm 5cm
5cm 5cm
13. .
5cm 5cm 5cm
5cm
14. 8 cm.
15. ( = ) = 120, = 3 cm. = . ) . ) .
3cm
3cm
232
1
t t t t t
1 ) ; ) , ; ) : 90 360 180 60 45 2 . . x + 10
3
A x + 20
0 2x + 10
3 62,8 cm. . ;A
4 12cm. .
233
2 1 ) ; ) : 72 3 6
t t t t t
140
2 x, y, z :A 120 100 50 x z y y x 60 z 0 40
3 . , 10m2 1 ;
10m
30m
50m
234
4 = 6 cm. 6 cm .
) ) . ) .
A 6cm 6cm 6cm
3
235
4
4
4.1 , , . (p, q, r).
p
. p , , p.
, . . , .p q p q
: . . 1 2 , :
239
, . 1//2. , . , . 1 2 .
2 1 p 2 p 1
2 1
q, : , q. , q. , q. , . q, , q. .
q
A
B
q
q
p
A
q
240
p q, p q.
p
4
q
6cm 3cm
1) .
3cm
4cm
, 2 = 2 + 2 2 = 62 + 42 2 = 36 + 16 2 = 52 = , 2=2+2 2=32+ 2=9+52 2=61 = 2) : ) ) ) 5cm 8cm
6cm
) , : 2 = 2 + 2 2 = 82 + 62 2 = 64 + 36 2 = 100 = = 10cm. ) : 2 = 2 + 2 2 = 52 + 102 2 = 25 + 100 2=125 = 11,18 cm.
241
) :
() (): 1) . 2) . 3) . 4) , , . 5) . 6) . 7) p, .
1. : ) ) ) .
3cm 4cm
2. .
242
12cm
3. , q =17 cm =12cm. =12 cm, .
4
q
4. : ) ) )
8cm 12cm
9cm
5. . . , , .
6m 5m 4m
6. . ( = 18cm, = 10cm, = 28cm)
4.2 , , . ,
243
. , . . , , ... , , ...
. = ( ) () . = + 2
, . . . , .
0
0
244
, : = 2 : = + 2 = 2 = 22
p
4
p
A
1) , =9 cm, =12cm 10cm.
9cm
12cm
10cm
: 2 = 2 + 2 2 = 92 + 122 2 = 81 + 144 2 = 225 = 15 cm : = ( ) () = (9+12+15) 10 = 36 10 = 360 cm2 = + 2 = 360 + 2 = 360 + 108 = 468 cm2.
245
2) . 5m 3m.
5cm 30cm 5cm
, : = 2 = 2 3,14 5 30 = 942cm2 = 0,0942m2 : = 5 3 = 15m2 : 15 : 0,0942 159 .
: 1) : ) : 5 : 6 : 7 ) : 8 : 6 : 9 ) : 12 : 9 : 6 2) 8cm 5cm. 9cm, : ) : : 117cm2 : 162cm2 : 234cm2 ) : : 314cm2 : 242cm2 : 157cm2 3) 10cm 10cm. ) : : 314cm2 : 942cm2 : 628cm2 ) : : 1256cm2 : 628cm2 : 1884cm2
246
1. 80cm, 40cm 30cm. 2. 25m , 15m 2,5m . : ) ) 25cm 0,30. 3. =192cm2 8cm. .
4
4. ( ) . , 1m2 15.0,4m
45cm 0,8m 60cm 1,2m 0,4m
5. 16cm 8cm. 6. 50,24cm 20cm. . 7. , 50 , 1,50m 0,20m 3 1m2. 8. , : ) 4m 3m. ) 37,68cm 12cm. ) 78,5 cm2 10cm.
247
9. 1000 1,2m 80cm. , 10% 1m2 1,5; 10. 30cm 60cm. ) . ) 10m 18m. .
4.3 , . (m3), 1m. : m3 dm3 cm3 mm3 1dm3 = 0,001m3 1cm3 = 0,000001m3 1mm3 = 0,000000001m3
(1L=1dm3) m (1ml=1cm3). . V= ( ) ()
248
.
V= ( ) () V= 2
4
1) 8cm 12cm. = 8cm, = 4cm. : V=2 = 3,14 42 12 = 602,88cm3. 2) 87,92 cm 10cm. L = 2 87,92 = 2 3,14 87,92 = 6,28 = =14cm. V = 2 = 3.14 142 10 = 6.154,4cm3. 3) = 6cm, = 8cm 20cm. : ) ) ) ) . ) : 2 = 2 + 2 2 = 62 + 82 2 = 36 + 64 2 = 100 = = 10cm. ,
249
, . ==10 =5cm. ) : V = 2 = 3,14 52 20 = 1570 cm3. ) : V = ( ) () V= ) : V = 1570 480 = 1090cm3.
1) , , . 25cm2 4cm 144cm3 16cm2 50cm 160dm3
2) , , . 25 cm2 8cm 294 cm3 49 cm2 4cm 314cm3
250
3) () (). ) ) ) V = 2
1. . 135cm3, : ) ) . 2. = 12cm = 16cm . : ) ) ) . 3. . = 2m, = 24cm, = 50cm 2m = 30cm. : ) 50cm =24cm ) 30cm ) .
4
4. ( = = 5cm) 4cm. 15cm, : ) ) . 5. : ) 30 cm 0,8m ) 942mm 0,4m. 6. 452,16cm2 24cm. . 7. .
251
70mm 100mm. , . 8. , 10cm. 9. 8m. , 3m 2,5dm.
4.4 O I I , . . . , , , K
. . . , , . , . , .
252
, , . .
: = ( ) ()
4
= + :
V=
( ) ()K
- , , .
1) 8cm. 12cm, . = = ( ) ()
(8 8) 12=384cm2.K
2) .
8cm 5cm 5cm
253
: 2 = 2 + 2 82 = 2 + 2,52 2 = 64 6,25 2 = 57,75 = : = ( ) () = (4 5) 7,6 = 76cm2.
: = + = 76 + 52 = 101cm2. : 2=2+2 2 = 52 + 52 2 = 50 = = =3,55cm.
: 2 = 2 + 2 82 = 2 + 3,552 2 = 64 12,6025 2 = 51,3975 = : V= ( ) ()= 52 7,2=60cm.
1) () (). ) . ) . ) . ) . ) .
254
) , .
4
1. 15cm 20cm. . 2. 12cm 10cm. . 3. 10cm 12cm. : ) ) ) . 4. . 384cm3 8cm 10cm 12cm 12cm 240cm2 10cm
5. 16cm 17cm. 6. 5cm. ( ). : ) ) , 4,1cm. 7. . .
255
8. 8cm 10cm. : ) ) ) ) . 9. = 12cm, , . .
= 8cm
10. : ) ) .
4.5 . . = . p , 0 . . . : 2 = 2 + 2 :
256
=
: = + = + 2 : V= V= 2
4
( ) ()
1) = 15cm 9cm. : V= 2 . 2 = 2 + 2 152 = 92 + 2 2 = 152 92 2 = 225 81 2 = 144 = = 12cm. : V = 3,14 122 9 = 1.356,48cm3.
2) 50,24cm2 5cm. : ) ) . ) : = 2 50,24 = 3,14 2 2 = 16 = 4cm. = = 3,14 4 5 = 62,8cm2 ) 2 = 2 + 2 2 = 2 2 2 = 52 42 2 = 25 16 2 = 9 = =3cm. : V= 2 = 3,14 42 3 = 50,24cm3.
257
3) : ) ) ) ) .
p
) : 2 = 2 + 2 = 2 = 36 = 6cm. p = 2 = 18 + 18
= 2 + 32 2 = 18 9 ) 2 = 2 + p2 2 = 9 = 3cm ) : Eo = + = p + p2 = 3,14 3 28,26cm2 = 28,26 ) : V= p2 = 3,14 32 3 = 28,26cm3. + 3,14 32 = 28,26 +
() (). ) 2 = 2 2. ) . ) V= 2. ) . ) . ) .
258
) . ) .
4
1. 8cm 6cm. 2. To 226,08 m2 = 9 m. .30
3. .p = 3cm
4. 18,84dm. = 5dm. : ) ) .50cm
5. . : ) ) .
5cm 20cm 25cm 20cm
6. . ) . ) 30 m2, 50
12cm
10cm
7. . 19,625m3 3 m. .
259
8.
-
9 cm. (dm3 ). 9. 20 cm . .20cm 20cm
20cm
20cm
10. 6 cm 10cm 15 cm . .
4.6 I I , p (,) . p p . (,). .
p
p
A
260
p
, , . , . . E = 4p2 : V = p3
4
1) = 10cm. . = 10cm, = 4p2 = 4 3,14 52 = 314cm2 V = p3 = 3,14 53 = 523,33cm3
2) 305,08 cm3. :
p3 = 729 p3 = 93 p = 9cm
261
: = 4p2 = 4 3,14 92 = 1017,36cm2. 3) 12cm . . = 12cm, : V = , : V = 123 = 1728cm3 : V = V V= 1728 904,32 = 823,68cm3. 4) 153,86 cm2. . = p2 : 153,86 = 3,14 2 2 = : V = 2 =49
() () 1) . 2) . 3) 4) , .
262
5) , . 6) , .
4
1. = 8cm. . 2. 50,24cm. : ) ) ) . 3. 2cm 3cm. 4. 113,04dm3. N . 5. 18m, m2 5. 6. 1256cm2. . 7. 6cm. 8. 30cm. . 9. = 25cm. : ) ) . 10. = 30cm . :
263
) ) ) 11. = 8cm . .
4.7 , . . 90 90 . () (S). . ( . ) , () (W). 180 180 . . ( ) . ( ) . . , (W) (), () (S).
264
1 1 ) . ) . ) . 2 : ) ) x 12cm 4cm 3cm
4
3 = 6cm = 8cm = 15cm. : ) ) . 4 15cm 16cm. : ) ) .
2 1 ) ; ; ) ; ) . ) . 2 7m 0,4m. , 150 . 3
265
502,4cm2 20cm. : ) ) ) . 4 20cm. .
266
-
11.1 1) 2 4 6 3 , , , t t t t t
1
2) ,
1. ) 5x + 7, x ) = 4x = x2, x ) 3 x, x ) 0,8 x, x ) = 2x + 2(x 5) = 4x 10, x ) x + 10, x
t t t t t
2. ) 9x 4y ) 22 27 ) 2 + 3 ) 3,1x 2,8y ) x 3. ) = 5x 12 + 6x + 8 + 4x = 15x 4 x = 3 = 15 ( 3) 4 = 49 ) = 7 14 6 6 + 5 = 20 + 5 = 7 = 1 = 7 20 ( 1) + 5 = 32 ) = 19 2 + 2 2 + x 12 = 3 + x + 7 x = 5 = 6 = 3 ( 6) + 5 + 7 = 30
269
~
4. ) = 5 10 6 + 9 + 7 = + 7 + = 8 = ( 8) + 7 = 15 ) = x + 6x 2y 4y + 12 + x = 6x 6y + 12 x y = = 6 (x y) + 12 = 6 + 12 = 1 + 12 = 13
) = 3 + x y + + 1 = x y + 4 x + y = 11 = ( 11) + 4 = 15 5. = 2 + 2 = 3 = = 1
= 3(
)(
)1
=
6. = 3x 4y + 5x 7y + 10 9x + 8y = x 3y + 10 = ( 10) 3( = 21 7. ) x = 20 x ) = x(20 x) = 20x x2 x = 10, ) + 10 y=
8. M = x (x ) (x ) = x x + x + = + x
1.2 1) ) 8 ) ) ) ) ) ) 6 ) iv i v ii ) 7 ) ) 69 ) ) 5 ) ) 5 )t t t t t
270
1. ) x = 4 ) x = 3 ) x = 2 ) x = 21 2. ) x = 4 ) y = 1 ) x = ) = 3. ) x [3 + (x x 3)] = 5 x=5 ) x ( 3x 1 5) = 2x 2 x + 3x + 1 + 5 = 2x 2 6x = 8 x=
t t t t t
1
) {2(x 4) 3(x + 1) + [10 2(x + 1) 60]} = 15(x + 1) ( 8 3x 3 + 10 2 60) = 15x + 15 8 + 3x + 3 10 + 2 + 60 = 15x + 15 3x 15x = 15 + 10 8 3 2 60 12x = 48 x=4 4. ) 2(3x + 2) = 3(4x 5) 6x + 4 = 12x 15 6x = 19 ) 21x + 63 15 = 10 40x 61x = 38
) 15x + 15 = 12x 8 27x = 23
271
~
5.
2(6x 4) = 7(1 + 3x) 14x 12x 8 = 7 + 21x 14x 5x = 15 x=3
3(x + 3) 6x + 30 = 2(2x + 2) 3x + 9 6x + 30 = 4x + 4 7x = 35 x=5
3x + 3 2x 30 = 6x 2 5x = 25 x= 5
20x 15 21x 14 = 2 x 140 x = 109
4x 2 5 = 4x + 4 11 0x = 0
6.
3 (3x + 3) + 2(2 2x) = 5x 5 9x + 9 + 4 4x = 5x 5 18x = 18 x=1
5(3y 1) + 5 = 2(y 2) + 5(2y + 5) 15y 5 + 5 = 2y 4 + 10y + 25 3y = 21 y=7
272
13(1 ) + 4(2 4) = 2( + 11) 3 3 + 8 16 = 2 + 22 3 = 35 = 3 + 2( + 3) = 5 + 6 3 + 2 + 6 = 5 + 6 =
7.
144 4 2x = 9 15 18x 12 4 144 8x = 135 18x 48 10x = 57 x = 5,7
4 6 4 + 5 = + 2 24 4 + 5 = + 2 6 = 27
273
~x 10 45 300 = 60x 1 59x = 354
x = 6 8.
2(6x 18) = 30 + 3x 12 12x 36 = 30 + 3x 12 9x = 54 x=6
x 3 9 = 27x + 40 26x = 52 x= 2
274
120 3(5x 1) 2(x + 5) = 144 3(3x + 1) 4(2x + 1) 120 15x + 3 2x 10 = 144 9x 3 8x 4 0x = 24
1
9.
15x 15(2x 1) = 10x + 6 2 15x 30x + 15 = 10x + 12 5x = 3
18 + 6 = 34 6 = 16 =
80x 8 = 66 11x 91x = 72
x=
275
~
7x 3 36x + 12 = 3(7x 15) 7x 3 36x + 12 = 21x 45 7x 36x 21x = 3 12 45 50x = 54
10. ) =
= 20
4 (2x 6) 20 = 5(3x 6) 8x 24 20 = 15x 30 7x = 14 x= 2
) =
276
4x (5x 12) = 12 x 4x 5x + 12 = 12 x x =
1
) =
2 x + 6x 10 = 3x + 18 2x 4x = 26 x=
11. ) (5x 7)(3x + 12) (x 2) = 0 5x 7 = 0 3x + 12 = 0 x= x= 4
x2=0 x=2
) ( x + 3)( 2x 13)(7x + 3) = 0 x + 3 = 0 2x 13 = 0 x=3 x=
7x + 3 = 0 x=
) (x 2) (x2 + 5) = 0 x 2 = 0 x2 + 5 = 0 x=2 x2 = 5
12. (1 x) + 3 = 2x + 5 + ) = 5 : 5(1 x) + 3 = 2x + 5 + 5 5 5x + 3 = 2x + 10 7x = 2 x= ) x = 3, (1 ( 3)) + 3 = 2( 3) + 5 +
277
~
+ 3 + 3 = 6 + 5 + 3 = 4 = ) = 2, 2(1 x) + 3 = 2x + 5 2 2 + 2x + 3 = 2x + 3 0x = 2 13. 7x + 5 = x + (7 )x = 5 ) : = 7 = 5 ) : = 7 = 5 14. ) = 2 ) = 5 ) = 1 ) = 1 15. : 16. : = 3y 5 = 3 y 4y = 8 y = 2 : 2 = 2 + 2 102 = 2 + 62 2 = 100 36 2 = 64 = 8 cm 5x 2 = 8 5x = 10 x = 10 3x + 10 = 4x 10 x = 20 x 20
1.31.t t t t t
278
2. i)
ii)
3. i)
ii)
1
11.
F = 73,4o
12.
279
~
1.4 1) . 2) .t t t t t
1. x , : 5x 5 = 4x + 9 x = 13 2. x , : x + (x + 2) + (x + 4) = 69 3x = 63 x = 21 : 21, 23, 25 3. x , : = 12
t t t t t
4x + 3x = 1092 7x = 1092 x = 156 4. x , x + 18 , 5x = 2x + 36 3x = 36 x = 12
12 18. 5. x , : 125 = x 17 + 6 17 x = 125 6 x=7
17 x = 119
280
6. x , 20 x, : 6x 3(20 x) = 75 6x 60 + 3x = 75 9x = 135 x = 15 7. x , 4x, x + 4x = 90 5x = 90 x = 18 18 4 18 = 72 8. x , 3x 2 (3x) = 6x. : 6x + 3x + x = 400 10x = 400 x = 40 6 40 = 240, 3 40 = 120 40. 9. x , x + 27. : x + (x + 27) + (x + 27) = 180 3x + 54 = 180 3x = 126 x = 42A x
1
B
x + 27 x + 27
10. x , : (x + 2) x = x2 + 50 x2 + 2x = x2 + 50 2x = 50 x = 25 cm 11. x , 60 x , : x(60 x) = (x 10)(60 x + 10) 100 60x x2 = (x 10) (70 x) 100 60x x2 = 70x x2 700 + 10x 100 20x = 800 x = 40 = 40 cm = 20 cm
281
~
12. x , 50 x, : x + 8 = 2(50 x + 8) x + 8 = 100 2x + 16 3x = 108 x = 36 36 14 . 13. x . 48 + x = 2 (18 + x) 48 + x = 36 + 2x x = 12 x = 12 14. x y , x 5 = 2(y 5) x = 2y 10 + 5 x = 2y 5 (x + 10) = y + 10
5x + 50 = 7y + 70
2y 5 = 10y 25 = 7y 20 3y = 45 y = 15 x = 2 15 5 = 25 25 15 . 15. x , 13 x . : 2,5x = 4(13 x) 2,5x = 52 4x 6,5x = 52 x = 8 8 2,5 = 20Km. 16. x , 80 (x + 1) = 100 x 80x + 80 = 100 x
282
20x = 80 x = 4 1.. 512 400 = 112Km . 17. x , :
1
x = 1.800 720
630
450
18. x , x 7. : 2x + 2(x 7) = 66 2x + 2x 14 = 66 4x = 80 x = 20 = 20cm = 13cm. 19. x , 3x. : 240 = 4x 10 40x = 240 x = 6cm 6cm 3 6 = 18cm
283
~
20. x , 48 x. : 2 x + 3(48 x) = 114 2x + 144 3x = 114 x = 30 x = 30 48 30 = 18 . 21. x , 12 x . : 1,5 x + 3 (12 x) = 30 1,5x + 36 3x = 36 1,5x = 6 x=4 4 12 4 = 8. 22. x, 2x 6 : 2x 6 = 3(x 6) 2x 6 = 3x 18 x = 12 12 24 . 23. 8 , 1 .
6 , . x , EK = 24
4x 3x = 24 x = 24 . 24. x , :
284
1x + 2x + 3x = 6 6x = 6 x = 1 . 25. x , 12(x + 0,5) = 14 x + 1 12x + 6 = 14 x + 1 2x = 5 x = 2,5 26. x ,
400 + 60 = 8x 460 = 8x x = 57,5% 27. x , :
24 .
x = 24
28. x , 16 x. : 5 x = 3 (16 x) 5x = 48 3x 8x = 48 x=6 6 .
285
~
1.5 1) ) x 2 10 )t t t t t
)
2) ) 3) . . ) ) i ii ii i
)
)
)
)
)
)
)
)
1. ) x 4 0 18 0 0 0 3 0
t t t t t
+ 4 +
) x < 18
) x >3
+
) x 6
+ 6 +
286
) x
2. ) x >1
0 1
+
) y > 3
3 0
+
1
) 6
0 6
+
) x 3
0 3
+
) x 0
0
+
3. ) x3
+
) x 3
0 3
+
) x 8
0 8
+
+ 0
)
)
+ 0
4. ) 3{x 5[ x (x + 2)]} < 15( x 3) 3[x 5( x x 2)] < 15x 45 3(x + 5x + 5x + 10)< 15x 45
287
~
3(11x + 10) < 15x 45 33x 30 < 15x 45 18x < 15 33x+15x x + 6 9x > 16 0
+
5. )
36x 4(10x 5) 3 (15x + 15) 24x 36x 40x + 20 45x + 45 24x 36x 40x 45x + 24x 45 20 25x 25 x 1
+ 1 0
)
2(2x + 30) 12x < 50 5x 3(x + 6) + 54 4x + 60 12x < 50 5x 3x 18 + 54 4x 12x + 5x + 3x < 50 18 + 54 60 0x < 26 x. 0 +
289
~
) 4 2( + 1) 8 3(3 )> 0 4 2 2 8 3 + > 0 3 > 13 0
+
)
18 + 3(x 2) + 2(x + 1) > 3(x + 5) 18 + 3x 6 + 2x + 2 > 3x + 15 2x > 1 0
+
)
14(2x 26) 21(3x + 5) 6 (2x 4) 210 28x 364 63x 105 12x 24 210 28x 63x 12x 364 + 105 24 210 47x 235 x55 0
+
6. )
290
2(x + 1) + 5x 10x + 20 (3x 4) 2x + 2 + 5x 10x + 20 3x + 4 0x 22 x. ) 5x + 1 3(3x 1) 2(5 2x) 5x + 1 9x + 3 10 4x 0x 6 . ) x + 2 4x > (3x 1) x 4x + 3x > 1 2 0x > 1 x. ) x 3 + 4(x 1) > 5(2 + x) x 3 + 4x 4 > 10 + 5x 0x > 17 . 7. ) x 7 < 2 5 x < 2 x < 9 x > 3 0 3 9 +
1
3 < x < 9 ) 2(x + 3) 3(x + 1) 2(x 1) 2x + 6 3x 3 2x + 2 x 1 1 0 8
3(x 3) + 7 < 2(x + 3) 3x 9 + 7 < 2x + 6 x 3x 24 2x + 3x 21 < 8 2x 1 6x 2 + 15x 40 > 3x 24
291
~
7x < 28 x 18 x >1+ 1 4
1 < x < 4 )
4x 8 + 14 < 3x 15 x < 21 21 0
9x 10 > 8x + 6 x > 16+ 16
) 3x 2 < 13 2(x 3) > 2 3x 5(x 1) 1 3x < 15 2x 6 > 2 3x 5x 5 1 x 4 2x 6 x>2 x3 + 0 2 3 5
3 x < 5
) x>3 6 3 0 2
x6
x2+
3 < x 2 8. ) 11 3x + 2 < 29 11 2 3x < 29 2 9 3x < 27 3xx> 3 ) 9 8x + 1 13 8 8x 12 1x
3 0 1
+
1
0 1
+
9. < 0 5( 2) 30 < 0 5 10 30 < 0 5 < 40 (4 2) 28 5 + 2 > 2 7 > 30
11. x , : 0,40 x > 10 x > 25 25 . 12. x . A x = 13 + 7 () . : 55 < x < 65 55 < 13 + 7 < 65 48 < 13 < 58 3,692 < < 4,46. = 4
x = 13 4 + 7 = 59. 13. x . 3x > 750
293
~
x > 250 x < 252 250 < x < 252 x = 251 251 . 14. x .
x + 16 + 51 + 72 + 76 + 20 > 18 14 x + 235 > 252 x > 17 15. x 2 , 100 + 0,5 x >150 + 0,3 x 0,5x 0,3x > 150 100 0,2x > 50 x > 250
1 1 . ( ) . ) ) ) 2 ) x = 3 . x > .
t t t t t
)
) x > 1
)
4 x
294
5x + 3x + x = 5 180 9x = 900 x = 100
1t t t t t
2 1 ) 2 . x = 7 3 . . = 10 >5 ) ) ) )
. = 2
4 26 .
295
22.1 t t t t t
2
1) ) , ) , ) , ) , ) , ) , ) , ) , ) , ) 2)
3) 4) 5) 6)
1.
t t t t t
2. ) 24 3. = =
) 38
)
) 19 3x 15 0 2 6x 0 x5 x 1/3
297
~4.
= 5x9
x 5 0 6x+54 0 x 5 x 9
5. x = 6,
y = 13,
z = 1,
= 25,
=8
6. ) x = 6 x = 6 ) x = x=
) ) x = 13 x = 13 7. = 2 8. = 40m 9. = 15cm E = 120cm2 10. = 17cm, = 4 17 = 68cm 11. = 24cm = 1008cm2 12. = 24cm = 26cm 13. x , : x2 8 = 14. = 3, 2x2 16 = x2 x2 = 16 x = 4 = 4, = 8, x=6
2.2 1) ) , 2) : ) , ) , ) , ) t t t t t
298
:
1.
t t t t t
2
2. 3.
1
0
1
2
3
4
5
6
7
4. ) x = ) x =
x= x=
)
) x = 1 x = 1
5. x cm . : x2 + x2 = 82 x2 = 32 x = x = 5,65cm
6. =
. = 18cm2
7. = 6,92cm E = 27,68 cm2 8. = 45 .
, = = 4cm : 2 = 42 + 42 2 = 32 = 5,65cm . =
45 4cm
299
~
. = 8= = 2,83cm
9. 2 = 62+122 2 = 180 = 13,41cm = 2 13,41 = 26,82cm 10. . = 16cm . 2 = 2 + 2 162 = 2 + 82 2 = 192 = = 13,85cm = 2 = 2 13,85 = 27,7cm E. = 11. :16cm 16cm
60 16cm 16cm
0
= 16cm 12. = 6cm . : 2 = 2 + 2 = 6,7cm = 6+6+9+6,7 = 27,7cm6m 18m 3m 6m
300
2.31. = 8,696m 2. x = 3cm (1 : 5cm) x= (2 : 5cm 4cm.t t t t t
2
3. 2 = 2 2 2 = 202-162 2 = 144 = 12cm AB2 = 2 + 2 2 = 362+122 2 = 1440 = 37,947cm 4. xcm , x+ = 12,07 2,41x = 12,07 x = 5cm = 20cm = 52 = 25cm2 5. 10 . 6. x2+(3x)2 = 302 x2+9x2 = 900 10x2 = 900 x2 = 90 = 5 x2 = 5 90 = 450cm2 7. 2 = 2 2 2 = 292 212 2 = 400 = 20cm . = 21 20 = 420cm2 8. 2 = 2 2 2 = 1296 = 36cm.
=21,6cm.
9. AB = 8 + 20 + 8 = 36cm 2 = 82 + 82 2 = 128 = =11,31cm 45 A
20cm 8cm 8cm
45 8cm
8cm
301
~
10. 2 = 22 + 42 2 = 20 = 2 = 22 + 42 2 = 20 =
2 = 22 + 62 2 = 40 = 2 = 2 + 2 . . 11. =50m = = 150 50 = 7.500m2 7,5 . 50 7,5 = 375 .
1 1 . ( ) . i) . ( ) 2 ) x 3 ) x 4 3 ) =8cm ii)
t t t t t
) =120cm2
4 = 15m =76m 12 76 = 912 .
2 1 ) . 7 ) . 4 ) . ) . 4 ) .
t t t t t
302
2 ) 125 ) 23 ) 153 ) 3 = = 4 x = 4cm, y = 6cm, =
2
303
33.1 1) ) y = 1,1x 2) ) y =x 4000 3) ) 4) ) = 20 + 5xt t t t t
3
1. x y 2. x y 3. x y 3 30 1 12 0 6 2 0 5 6 2 5 1 1 0 3 1 1 2 5 1 2 0 1 1 3 4 4
t t t t t
4. y = 1,19 x 5. y = 200 + 0.5 x x y 100 250 200 300 300 350 500 450
305
~
6. E = 4x2 7. ) y = 25 x. x = 8 y = 17 ) x = 15 y = 16
8. ) y = 1,05 x x y 900 945 1000 1400 1500 1050 1470 1575
9. y = 2x 3 x y 5 7 2 7 2 1 6 9
10. ) S = 3 450 = 1350 ) S =450 t ) 2.250 = 450 t t = 5 .
3.2 1) (3,4), 2) (3, 7) (5, 6) (1, 4) (8, 10) xx (3, 7) (5, 6) (1, +4) (8, 10) yy (3, 7) (5, 6) (1, 4) (8, 10) 0 (3, 7) (5, 6) (1, 4) (8, 10)t t t t t
(0,3),
(4,3),
(25),
(1,2)
306
3) ) < 4) ) : = 90
3
) : = ) : =2 ) : = 5 .. 5) ) y = 4, ) y = 2, ) y = 1, ) y = 0, ) y = 2, ) y = 3
1. y 3 (4,2 2 (0,2) 1 x 4 3 2 1 0 1 2 2,7 3 4 y 1 2 3 3,5 4
t t t t t
Z(5,0) 5
x 6
E(3,5, 2,7)
2. ) (7,2)
) (7,2)
307
~3.
) (7,2)
) 2 = 2 + 2 20 = 10 + 10 20 = 20 )
4. A (5,2) (-2,4) (-1,0) 5.
xx 2 4 0
yy 5 2 1
6. y = 2x2 x y 2 8 1 2 0 0 1 2 2 8
308
)
y 8 7 6 5 4 3 2 1
)
y 8 7 6 5 4 3 2 1
3
x
2 1 1 y
x 1 2
x
2 1 1 y
x 1 2
x
x
7. (3,15) y = 2x2 + : 15 = 2(3)2 + 15 = 18 + =3 3
8. , 500 , 500 : 400= 1,25 1 1 .4 (3,4)
9. 90 ) 86 80 7768 70 60 50 41 32 20 10
>
>
> > >
>
>
>> 5
10 15 20 25 30
> > > > c
) 10 C F 50 ) 77 F C 25
309
~
10. ) y(Kg)90 81 76,5 72 67,5 63 > > > > > > > > > > > 160 170 180 190 200 210 175 185 > > >
) 175 cm 67,5 ) 76,5 185 cm.x(cm)
3.3 1) ) x y 3 4 6 8 9 12 12 16 )t t t t t
2) : 3) : y= 3x 4) ) 2, ) 1, ) 1, ) 2, ) 1
y
t t t t t
1. )
x y
2 3
4 6
6 9
8 12
)3
310
)
x y
2
x
2. x y
y = 3x 0 0 1 3 x y
y = 5x 0 0y 5 4 3 2 1 x 3 2 1 1 2 3 4 5 y y = 5x y = 2x x 1 2 3 4 y = 3x
y = 2x 1 x y 0 0 1 2
5
3
3.y y = 2x 2 1 x 3 2 1 1 2 y x 1 2 3
y = 2x x y 0 0 1 2 x y 0 0 2 1
.
4. y=x, 6 = (4) = 5. y = 6. 1 : y = 2 : y = y =
311
~
7. t S
S=8t 0 0 1 8
S () S=8t
24 16 8 1 2 3
t ()
8. y= x (3,6) : 6 = 3 = 2 = 2 9. ) y=1,05 x ) x = 850 y = 1,05 850=892,5 ) y = 945 945=1,05 x x = 900 10. ) y =5x ) 52,5 ) 9 11. ) ) =
3.4 1) : y = 7x + 3 2) : (3,0) 3) : (0,7) 4) : 3x 2y = 6 5) ) : ) : (6,0) ) : (0,2)t t t t t
312
1.5 4 3 2 1 x 3 2 1 1 2 3 4 5 y x 1 2 3 4 y 2: y = 2x+3 1: y = 2x 3: y = 2x 2
t t t t t
3
1: y = 2x
x y x y x y
0 0 0 3 0 2
1 2
2: y = 2x + 3
0 1 0
3: y = 2x 2
2. )y 3 2 1 x 1 2 y y =5x+3 1 3 x
)y 3 2 1 x 3 2 1 1 2 y (x 0) x 1 2 3
(x )
313
~
)18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 x 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 (3 x 3
y
x 1 2 3
y
314
3. 1: y = (5 2)x + 3 2: y = ( + 2)x 9 , : 5 2 = + 2 4 = 4 =1
4. : 5. ) (2,1) (2,4)
3
) (2,1) 3x + 4y = 10 : 3(2) + 4 1 = 10 3x + 4y = 10 (2,4) 3x + 4y = 10 : 3 2 + 4 4 = 10 3x + 4y = 10 6. 4x 3y = 12 xx (3,0) yy (0,4). x 0 (3,0) (0,4) y y
x
7. )
2x + y = 4 x y 0 4 2 0x 0 4
y
2
x
) = 2 ) 8. ) (5,1) ) = (2,1) (2,3) (5,3)
y
315
~
9. y = 5x7, , = 5. y = 2x + y = 5x + . (3,1) ; 1 = 5(3) + = 16. y = 5x + 16. 10. ) y = 0,05x + 800 ) y = 1200 : 1200 = 0,05 x + 800 0,05x = 400 x = 8.000 y )1600 1200 800 400 x y 1000 2000 3000 4000 5000 6000 7000 8000 9000 x y = 0,05x + 800
11. ) y = 80 x 1200
)400 x 400 800 1200
y y = 80 x 1200 x
5
10
15
20
25
y
12. ) t = 0 S=600 80 0 = 600Km ) S = 0 0 = 600 80 t t = 7, 5 ) S600
0
7,5
t
316
13. ) (2,2) : 2 = (3 + 1) 2 + 2 = 6 + 2 + 2 6 = 6 = 1 ) = 1 : y = 2x + 2 x y 0 2 1 0x 2 1 3 2 1 1 2 y x 1 2 3 y
3
14. () y = 5x + 7 2 + 1 = 5 = 3 (1,6) : 6 = [2 (3) + 1] (1) + 3 2 6 = 5 (1) + 3 2 3 = 3 =1 = 3 = 1
3.5 1) ), ), ) 2) ) , ) , ) , ) t t t t t
1. ) x y 6 3 1 1 3 1 3 3 1 6
t t t t t
317
~
y
)3 2 x 6 5 4 3 2 1 1 1 1 2 3 y 2 3 4 5 6 x
2. x y 5 4 2 2 5 4 2,5 5 2y 9 8 7 6 5 4 3 2 1 x 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 x
x y
5 2
4 2,5
2 5
2
4
5
2 2,5 5
5 2,5 2
318
y
3.
x y x y x y
4
2 1
1 2 1 4 3 4
1 2 1 4 3 4
2 1 2 2 4 3
4
3
4 1 6 2
2 2 4 3
4 1 6 2
y
x 0
x
y
4. )
:
3 = 25 ) = 1
2 = 2
=1
319
~
y
x 0
x
y
5. 12 (20 5) = 10 x
x = 18
6. 150 (10 4) = (150 x) 9 900 = 1350 9x 9x = 450 x = 50 .()
7. ) = ) 80 3 60 4 40 6 20 12
12 8 6 4 3 0 10 20 30 40 50 60 70 80 ()
8. ) ) x = 4 ) y = 6 6= 12 x = 8 .
320
1 1 ), ) ( ) 2 y =5x2 7 3 (1,3) x y 0 7 2 13 1 2 4 73 5 118
t t t t t
3
(7,9)
4 (2,2) y = : = 0 x y 4 8 3 2 2y 1 x 5 4 3 2 1 1 2 3 4 5 6 7 8 9 y 1 2 3 4 5 x
x2 +
=0
1
0 0
1
2 2
3
4 8
321
~
2 1 ), ) ( )
t t t t t
2 y = x (3,15) : 15 = (3) = 5 y = 5x 3 ) xx (3,0) yy (0,7) ) =
) 4 , : 6 10 = x 15 60 = 15 x x = 4
3 1 . . ( ) 2 ) y = 1,08 x ) x = 120 : y = 1,08 120 = 129,6 ) y = 216 : 216 = 1,08 x x = 200
t t t t t
322
3 ) y = x
:
3
)
4 y = x + (0,3), 3 = + 12 = 3 + 0 y = 4x = =4 (3,12),
323
44.1 1. ) 2. ) 3. )t t t t t
4
t t t t t
1. ) 130 ) 45 ) 167 ) 18 ) 8 2. ) 3,5 ) 24 ) 56 ) 208 ) 450 3. ) 40% ) 400% ) 100% ) 80% 4. ) 165 5. 40% 60% . 6. "" : 0,3 30%
"" :
0,45 45%
"" :
0,25 25%
7. ) :
0,7 70% 0,35 35%
) :
8. . , . 9. . 200 . .
325
~
4.2 1. ) . 100 ) . 20 ) . 65 ) . 65 ) . 0t t t t t
) . 10%
t t t t t
1. ) : 4.000 + 4.500 + 3.5 00+ 7.000 + 6.000 = 25.000 . ) 2005 : )( ) 8000 7000 6000 5000 4000 3000 2000 1000 2002 2003 2004 () 2005 2006
0,28 28%.
2. ) : :
:
126 72 54 108
:
326
:
) : 3.(gr) 180 250 200 150 100 40 50 A
()
4
: : : :
( )
75 12520
50 90
: :
4. ) 6 + 32 = 38 ) 10 2 ) 10 + 2 = 12 2 ) 64%
327
~
)
36 32 28 24 20 16 12 10 8 6 4 2 0 1 2 3
) 0 : 1 :
1
0
2 :
43,2 230,4 14,4 3 72
2
3 :
5. : 90 : 270
4.3 1) ) . 20 ) . 5 ) . ) .t t t t t
) .
328
2)
40 16 32 48 24 160
. (%) 25 10 20 30 15 100
4
1. 1 2 3 4 5 6 2 4 6 5 2 1 20
t t t t t
. (%) 10 20 30 25 10 5 100
2.
6 2 8 10 8 6 40
. (%) 15 5 20 25 20 15 100
329
~
3. )
. . (%) 26 27 28 29 30 4 6 3 2 1 16 25 37,5 18,75 12,5 6,25 100
) 3 + 2 + 1 = 6 28 ) 4 + 6 + 3 = 13 28 4. ) 0 1 2 3 4 5 6 )(. )
I II III IIII IIII III I
1 2 4 4 5 3 1 20
. (%) 5 10 20 20 25 15 5 100
5 4 3 2 1 0 1 2 3 4 5 5
) 1+2+4+4=11 3 . ) 20% 5 .
330
5. ) . % 30 15 5 50 60 30 10 100 216 108 36 360
4
)30 (. ) 25 20 15 10 5 0 216 108 36
6. + 2 + 126 + 3 + 126 + 72 = 360 3 = 162 = 54 40 60 70 30 200 . (%) 20 30 35 15 100 72 108 126 54 360
331
~
7.
. 0 1 2 3 4
12 10 18 7 3 50
. (%) 24 20 36 14 6 100 56%
) : 28
) : 40
80%
) : 10
20%
) 2 3 : 25
50%
4.4 1) 1. . 4 2. . 14 3. . 8 4. . 20%t t t t t
1. ) = 18 3 = 15 =
t t t t t
332
) 36 69 9 12 12 15 15 18 )10 8 6 4 2 0 3 6 9 12 15 18
IIII IIII III IIII III IIII IIII IIII IIII
4 8 8 10 10 40
. (%) 10 20 20 25 25 100
4
2. ) 33 30 22 15 0 20 30 40 50 60
)
. .
. (%)
20 30 30 40 40 50 50 60
22 30 33 15 100
22 30 33 15 100
) 33% + 15% = 48% 40 .
333
~
3. ) () 50 55 55 60 60 65 65 70 ) 8 . % 7 6 4 3 2 0 50 55 60 65 70
III IIII IIII II IIII I
3 4 7 6 2040 35 30 20 15 10 0
. (%) 15 20 35 30 100
50 55 60 65 70
4. ) () . % 86 () 80 64 60 40 30 20 0 0 50 100 150 200
0 50 15% 30
50 100 43% 86
100 150 32% 64
150 200 10% 20
) 64 + 20 = 84 100 . ) 30 + 86 + 64 = 180 150 .
334
5. ) 8 + 12 + 14 + 10 + 6 = 50 ) 12% 16
4
)
54% .
4.5 1) . 6 4) . 24 2) . 567 3) ) . 25 ) . 27 ) . 23 5) . 360 fit t t t t
1. 2. 3. x , 2x,
t t t t t
35 = 29 + 3x 3x = 6 x = 2 2 4. 4. ) ) : 172, 175, 177, 183, 189, 190, 193, 195
)
335
~
5. 02 24 46 68 8 10 1 3 5 7 9 . 8 12 10 6 4 40 ( ) () 8 36 50 42 36 172
6. :
7. : 1.150 (v + 5) = 1.200v + 5.000 1.150v + 5.750 = 1.200v + 5000 50v = 750 = 15 . 8. y 2 = 14 y = 16
126 = 19x + 88 19x = 38
x=2
336
9. ) 25 35 45 55 ( . . % ) () 15 20 10 5 50 30 40 20 10 100 375 700 450 275 1800
4
20 30 30 40 40 50 50 60
IIII IIII IIII IIII IIII IIII IIII IIII IIII IIII
)
1 1 ), ) ( ) 2 ) 56,25% .
t t t t t
) 3 )
33,33% .
13 14 15 16 17 18 19 20
2 1 3 3 4 2 3 2 20
. . % 10 5 15 15 20 10 15 10 100
337
(. )
~
)5 4 3 2 1 13 14 15 16 17 18 19 19
()
) 2 + 1 + 3 + 3 = 9 16. ) 20% + 10% + 15% + 10% = 55% 17. 4 ) ) : 168, 169, 170, 171, 172, 175, 175, 177, 180, 183
2 1 ), ) ( ) 2 3 )
t t t t t
)
+ 9 + 4 = 19 5 .
338
4 ) : f1 + f2 + 0,3 + f4 = 1 f2 = 2f1 f1 = 2f4 2f4 + 4f4 + 0,3 + f4 = 1 7f4 = 0,7 f4 = 0,1. f1 = 0,2 f2 = 0,4 ) 167 0,2 + 173 0,4 + 179 0,3 + 185 0,1 = = 33,4 + 69,2 + 53,7 + 18,5 =174,8cm.
4
339
11.11. ) 1= 36 ) 1=18 ) 1= 9 2. 2=42 2=21 2=10,5t t t t t
1
3.
1.2 1) 1. B. 450mm2 2. B. 51000cm2 3. . 0,00039dm2 4. . 0,072m2 2) 1. 2,7Km2 = 2.700.000m2 = 2.700 2. 46m2 = 4.600dm2 = 460.000cm2 = 46.000.000mm2 3. 528dm2 = 5,28m2 = 0,00528 4. 7903mm2 = 79,03cm2 = 0,7903dm2 0,007903m2t t t t t
1. 25m2 = 250.000cm2 2,16 Km2 = 21.600.000.000cm2
t t t t t
341
B~
143dm2 = 14.300cm2 5779mm2 = 57,79cm2 712m2 = 7.120.000cm2 2. 498cm2 = 0,0498m2 111dm2 = 1,11m2 12,7Km2 = 12.700.000m2 13.534mm2 = 0,013534m2 607dm2 = 6,07m2 3. 456m2 = 456.000.000mm2 82,7dm2 = 827.000mm2 0,571cm2 = 57,1mm2 0,0025m2 = 2.500mm2 4. 914m2 = 0,000914Km2 4832dm2 = 0,00004832Km2 17075m2 = 0,017075Km2 103 = 0,103Km 5. 72564m = 72,564 3,4Km = 3400 137920dm = 1,3792 45m = 0,045 6. m 98,3 7,56 3,2103 0,738019 dm 9.830 756 321,03 73,8019 cm 983.000 75.600 32103 7380,19 mm 98.300.000 7.560.000 3.210.300 738.019
342
1.3 1) 1. 2. 2) 1. 2. 3. 3. 4. 4. 5. 5. 6. 7. 8. t t t t t
1
1. . . = : 144 = 122 = = 12cm : 4 12 = 48cm.
t t t t t
2. xm , 2x m . : 2 x + 2 2x = 162 2x + 4x = 162 6x = 162 x =27 27m 2 27 = 54m. . = () () = 54 27 = 1458m. 3. , : . = 2 16 25 = 800cm 127 : 127 800 = 101.600cm. 4. . = 46 20 = 920m 300 920 = 276.000g = 276Kg. 2 276 = 552 . 5. . = 8 7 = 56cm = . = 2 8 + 2 7 = 30cm = . . 2 10 + 2 x = 30 2x = 10 x= 5cm 10 1 = 56 5 2 = 56 1 = 5,6cm 2 = 11,2cm. 6. x , 2x :
343
B~
3x 6 = 72
3x = 12
x=4
= 4cm, = 8cm. 7. . () = 18 = 324cm () = 324 = 81cm, (AKB) = 18 x = 162 E() = 324 = 54cm, () = 18 y = 108 81 = x = 9cm
54 = y = 6cm
() = () () () = 324 81 54 =189cm. 8. () = () 4 () = 32 18 4 9. ) : = 1 = 2 () = ) = = () = = = 576 288 = 288cm.
3 = 1 = 2 () = () = () () = 10. ) x = 81 ) 5 x = 60 ) ) () x = 9cm x = 12cm x = 6cm x = 64 x = 8cm
= 24 = 32
344
11. ) 24,5cm 12. () =
) 300cm
) 228cm
) 70cm
) 169cm
1
() = 4 30 = 120m 975 120 = 855m 13. () = 1024 = 1024 = 32cm () = 676 = 676 = 26cm E() = ( = = 26cm = = 32 26 = 6cm). 14. () = () () E() = 80 = 6400 300 300 = 5.800m
5,8 : 5,8 12.000 = 69.600 . 15. ) = = 9 + 2,25 = 11,25cm
) () = () () () ()= =85 = 40 6,25 5 5,25 = 23,5cm 16. () = () () E
=
() = () () () = ()
, . () = () + ()
345
B~
1.4 1) 1. (x = 6cm) 2. A(x = 13cm) 3. (x = 24cm) 2) 1. 2. 3. 4. 5. t t t t t
1. ) 26 =242 + 10 676 = 576 + 100 676 = 676 ) 29=20 + 21 841 = 400 + 441 841 = 841
t t t t t
) 6 = 5 + 4 36 = 25 + 16 36 = 41
2. =12 + 9 = 144 + 81 = 225 = =15cm = = 15 =225cm 3. .. 8 = + 4 64 = + 16 = 64 16 = 48 = = 48cm
9cm
12cm
8cm 4cm
8cm
4cm
4. ) .. . : = 40 + 30 = 2500 = 50m = 4 50 = 200m ) 15 200 = 3000
40m 30m 0 30m 40m
346
5. = 21 + 20 = 441 + 400 = 841 = 29Km 6. AB = 8 + 15 AB = 64 + 225 AB = 289 AB = 17 AB = 17cm6cm8cm 8cm
1
15cm 6cm
7. .., 150 = 75 + 130 22500 = 5625 + 16900 22500 = 22525 . 8. ) = 10 8 = 100 64 = 36 = 6 = 21 6 = 15 x = 8 + 15 x = 289 x= 17 ) .. : = + 10 = + 8 = 100 64 = 36 = 6 = + = 6+9 = 15 x = + x= 15 + 8 x = 289 x = 17 ) .. : = + 10 = + 8 = 36 = 6 = + 17 = 8 + = 289 64 = 225 = 15 x = 6 + 15 = 21 x 10 8 21 x 10 9
8
x 17
8
10
347
B~
9. ) = =
= 40cm 40cm 40cm
) .. : = + 40 = + 24 1600 = + 574 = 1600 574 = 1024 = 32cm ) () = ) () = = 768cm 768 =
24cm 24cm 48cm
40 = 1536 = 38,4cm
10. = xcm, = 8 x cm .. : = + = 9 + x = 81 + x (1) .. : = + = 7 + (8 x) = 49 + (8 x) (2) = (1) : 81 + x = 49 + (8x) 81 + x = 49 + (8x) (8x) 81 +x = 49 + 64 + 8x 8x + x 16x = 32 x = 2cm. = 2cm. 11. .. : = + 10 = + 8 = 36 = 6cm = =3cm .. : = 8 + 3 = 73 = = 73cm 12. () = 136 = .. . EBZ : = +
10cm
8cm
348
136 = 6 + x x = 100 x = 10cm 10 + 6 = 16cm () = 16 = 256cm.
1
1 1 . . ( ) 2 = 20cm . = 20 14 = 280cm 3 ) ) ) 4 ) 16cm
t t t t t
) = 20cm
) 240cm
) 1 = 15cm
2 1 . ( ) . ) x = 5cm ) x = 7cm
t t t t t
2 27 15 = 12cm : 1 = = 20cm 2 = = 25cm
3 ) (7 x2)+(6x4) + 4x = 96 17x = 102 x=6
349
B~
: = 40cm, KM = 32cm = 24cm ) = + 40 = 32 + 24 1600 = 1600 .., . KM 4 . ..: = + 17 = + 8 = 289 64 = 225 = 15cm =
17cm 8cm
10cm
17cm
10cm 26cm
8cm
= 270cm
350
22.1 1) 1. 2) 1. 2. 2. 3. 3. 4. 4. 5. 6. 7. t t t t t
2
1. ) 40 = ) 50 = ) 48 = ) 60 = 0,839 = 1,192 = 1,111 = 1,732 = x = 0,839 5 x= x=
t t t t t
x = 4,195cm x = 5 cm
x = 1,732 15 x = 25,98cmy
2. x OA = 5cm oy = 3cm. =
B 3cm 0 5cm A x
3. 20 = 50 =
0,364 = 1,192 =
=
13,736 cm.
16,37 cm
x = + = 5 + 16,37 = 21,37cm.
351
B~
4. 25 = 42 =
0,466 = 0,9 =
= =
25,75m. 13,33m.
= = 25,75 13,33 = 12,42m.
5. 23 = 41 =
0,424 = 0,869 =
= 0,424 30 = 12,72m. = 0,869 30 = 26,07m.
x = = 26,07 12,72 = 13,35m. 6. = = = 0,08. 0,08 = A 350m 8%
B
= 0,08 350 = 28m. 7.
A
4cm
1,95 cm. 0,487 = =
64 B
26 =
8,21 cm.
= 1,95 + 8,21 = 10,16 cm. 8. = 2
= 2 , .
352
x
y
9. = 2x + 2y 19 = 2x + 2y 9,5 = x + y (1)
y 42 A x
2
B
42 =
0,9 =
y = 0,9 x
(2)
(1) (2) 9,5 = x + 0,9 x 9,5 = 1,9x x = x = 5 cm. y = 0,9 5 = 4,5 cm. = x y = 5 4,5 = 22,5 cm2 10. =
=
}) )
= 2 .
2.2 1) ) 2) : 3) ), ), ), ) . 4) ) , ) , ) , ) , ) , ) 5) ) 2 + 2 = 1 ) 2 + 2 = 1 ) = ) = ) = ) t t t t t
t t t t t
1. ) .. 2 = 2 2 2 = 2,52 1,52 2 = 6,25 2,25 2 = 4 = 2 cm.
353
B~
) 2 = 212 + 202 2 = 841 = 29cm
) 2 =202 162 2 = 144 = 12cm
2. 2 + 2 =1
2 =
=
=
=
3. 2 + 2 = 1 =
=
354
4. ) 5 3 > 2 3 > 3 < 1 ) < 1 7 < 7 7 + 4 < 11 < 1 4 < 4 ) 6 + 3 < 9 3 < 3 < 1 ) < 1 2 < 2 2 + 3 < 5 < 1 3 >
>
>
> B
>
}
(+) >
11. ), )F2 = 20N F F1 = 15N
)
2
12. ) ) ) (1) (2)
(1) (2)
2.7 1) : 8 2) : 17t t t t t
1.
t t t t t
367
B~
2.
,
3.
4.
5.
35 = 320 0,819 = 262,08
1 1 ), ) () 2 ) 30 =
t t t t t
x = 4cm
) 45 =
x = 3 cm
) 60 = 3
x = 15 cm
368
=
= 0,15 520 = 78m
4
2
2 1 ), ) () ) i) 37 < 43 ii) 56 > 16 iii) 17 > 52 iv) 89 > 1 v) 40 = 50 2 30 = x = 75cm
t t t t t
30 =
45 =
3 = = 700m
.. 2 = 2 + 2 2 = 7002 + 702 2 = 490000 + 4900 2 = 494.900
369
B~
4
3 1 () 2 5 4 = 0
t t t t t
2 + 2 = 1
+ 2 = 1
+ 2 = 1 2 = 1 2 =
=
=
3 ) .. : 2 = 52 + 122 2 = 25 + 144 2 = 169 = 13cm
A 5cm B O 12cm
370
)
2
) = 5cm . , 4
371
33.1 1) ) : 55, ) : 110, ) : 35 2) ) . 40, ) . 120, ) . 115, ) . 65 3) ) , ) , ) , ) t t t t t
3
1.
t t t t t
2. 3. 6x 10+ 2x +30 = 180 8x = 160 x = 20 4. 5. 6. 7. 8.
373
B~
9. 10.
11. . 12. , .
3.2 1. . 36 2. . 120 3. . 24 4. . 30 5. . 3 6. . 135 7. . 72 8. . 90 9. ) ) ) ) t t t t t
1. 4 5 8 12 90 108 135 150
t t t t t
90 72 45 30
374
2.
120 160 108 140
30 20 72 40
3
3.
= 12
4.
v =
O 40 B
5. (O,) . 6. ) v = ( )
)
= 180 140 = 40 v =
=9
.
375
B~
B
7. . .
8. 5 (,) 72. . 72. . 9. = 18 +72 = 90 .72 72
1 0 72 1
72 B
72 72
72
36
36 B 18 72 18 72
72
72
3.3 1. o o L 6cm 37,68cm 0,8m 5,024m 8cm 50,24cmt t t t t
18cm 113,04m
376
2. ) ,
) ,
)
1. L = 56,52 cm 2. p = 13 cm 3. L1 L2 = 26 21 2 2 = 26 (1 2) = 26 1 2 = 1 2 =
t t t t t
3
1 2 4,14cm.
4. ) 1 2 = 8 2 1 22 = 8 1 2 = 4cm ) L1 L2 = 1 2 = ( 1 2) = 3,14 8 = 25,12cm 5. L = 3,14 68 = 213,52cm = 2,1352m . 3.500 2,1352 = 7.473,2m = 7,4732m. 6. .. : = + = 12 + 16 = 400 = 20cm L = 3,14 20 = 62,8cm. 7. =2 = 2 45 = 90o . AOB . .. : = + 2 2 2 = + 2 = 18 L = 3,14 2 3=18,84cm.
2 = 9
= 3cm.
8. 24 24 . : L = 2 3,14 1,2 = 7,536cm L = 2 3,14 2 = 12,56cm 9. L = 3,14 70 = 219,8m 2 110 = 220m = 220.000m 220.000 : 219,8 1000 .
377
B~
3.4 1. 30 60 90 120 135 150 240 ) : 2L 2. ) : = 60t t t t t
1. 45 30 75 48
t t t t t
225
330
2. ) L = 2 50,24 = 2 3,14 =
= 8 cm.
378
) L =
= 12,56 cm.
3.
L
4cm 30o
5m 45o
4cm 270
3
2,09cm 3,925m
6cm
4.
62,8 cm.
5.
12 cm.
6. 6(x-2) + 3 = 4x 1
6x 12 + 3 = 4x 1 2x = 8
x = 4cm. O = 4cm = 2cm
7. : L
=
= 6,28cm
L
=
= 3,14 cm
L
= +L
= 9,42 cm +L = 6,28 + 3,14 + 9,42 = 18,84cm
=L
8. L =
= 7,85 cm
L = 3 (2 3,14 8) = 150,72 cm 7,85 + 150,72 = 158,57 cm.
379
B~
3.5 1) L 2cm 12,56cm 12,56cm2 4cm 25,12cm 7cm 43,96cmt t t t t
50,24cm2 153,86cm2
2) : 4 3) : = 4) : = L
1. = : 1256 = 2 = = 400 = 20 cm
t t t t t
L = 2 3,14 20 = 125,6 cm 2. L = 2 : 81,64 = 2 3,14 = 13 m = = 3,14 13 = 530,66 m. 3. 1 = 1 = 3,14 3 = 28,26 cm, 2 2 = 8 1 = 8 28,26 = 226,08 cm 2 = 2 226,08 = 3,14 2 2= 72 2 = 4. E. = 3,14 (52 32) = 3,14 16 = 50,24 cm. 5. .. = + = 3 + 4 = 25 = 5 cm L = 2 3,14 5 = 31,4 cm = 5 = 3,14 25 = 78,5 cm.
380
6. . = 32 = . . .. = + = + = 32 + 32 = 64 = 8cm = 4cm L = 8 3,14 = 25,12cm = 3,14 4 = 50,24cm.
0 B
3
7. = 16 cm = 8cm L= 16 3,14 = 50,24cm = 3,14 8 = 200,96cm. 8. = : 615,44 = 3,14 = 196 = 14cm 2 = 2 14 = 28cm = 4 28 = 112cm. 9. : = (R ) = R = = = 3,14 (5 ) = 157cm 157 = R 157 3,14 R = 314 R = 100 R = 10cm. 10. = = 2 = 2 3,14 = 3,14 4 = 8 =
3.6 1) : 8 (cm) 2) : 60 3) : 2 4) ) , ) ,t t t t t
) ,
) ,
)
381
B~
1. = 36 2. .. = 84,78 =
t t t t t
= 81 = 9cm L = 2 = 2 3,14 9 = 56,52cm. 3. = 1519,76 = 3,14 = 484 = 22cm .. = 4. L = = 303,952cm
= 4cm 4,18cm
.. =
5. )
=
= 75,36cm
) .. OAB: 122 = 2 + 36 144 = 2 +36 2 = 108 = = = 75,36 = 75,36 62,35 = 13,01cm
) = .. 7. =
= 104,66cm
8. () =
=
=
382
= 75,36 13,08 = 62,28 cm
9. . =
= 39,25 14,13 6,28 = 18,84cm
3
. = 3,14 5 + 3,14 3 + 3,14 2 = 3,14 (5 + 3 + 2) = 3,14 10 = 31,4cm 10. .. . = + = = 10cm = . = . 2 .. . = = 5cm.
= 50 39,25 = 10,75cm.
11. .. . = + 10 = + 5 = 75 = . = 2 ..
= 8,66cm
12. AB ( = = 45 = 14,13cm
= 90 = )
=
= 18cm2
. = 18 14,13 = 3,87cm .. : = + = 6 + 6 = 72 =
8,48cm
383
B~
L
=
= 4,71cm = 6 + (8,48 6) + 4,71 = 13,19cm
. = + +L
13. ) H M .. . (A2 = 2 + 2 2 = (10 = 400 = )2 + (10 )2 = 10cm
= 20cm
R = ) = 3,14 2
= 157cm
= 3,14 10 = 314cm ) = 314 157 = 157cm 14. . = 10 4 = 100 78,5 = 21,5cm
15. . = 2 + 2 = 8 = 64cm = = 25,12cm
=
= 33,49 27,71 = 5,78cm . = 64 2 25,12 + 2 5,78 = 25,32cm 16. )
) . =
.. 7,65 4,71 = 2,94cm
384
1 1 ), ) () ) 180 90 360 45 2 2x + 10 = 90 x = 40. = 50 = 130. = 85,
t t t t t
3
= 60, = 90,
= 120, = 95 = 90.
3 . = 4 62,8 = 4 = 15,7 cm = = 15,7 = 246,49 cm L = 2 62,8 = 2 3,14 = 10 cm = = 3,14 10 = 314 cm . 4 . = 4 12 +2
= 48 + 37,68 = 85,68cm
. = 2 ( ) = 2
= 61,92 cm.
2 1 ) () ) v 3 60 120 5 108 72 6 120 60
t t t t t
9 140 40
385
B~
2 ) ) = 110, = 20, = 50, = 25, = 60 = 135 = 70 = 45
3 . = 50 30 3,14 5 = 1500 78,5 = 1421,5cm 1421,5 : 10 = 142,15 4 ) L =L =L = +L +L = 6,28cm = 3 6,28 = 18,84cm
) = L
= .
+ 3 =
= 15,3 + 3(18,84 15,3) = 15,3 + 3 3,54 = 25,92 cm ) . = 25,92 15,3 = 10,62 cm.
386
44.1 1) , 2) , 3) , 4) , 5) , 6) , 7) t t t t t t t t t t
4
1. ) , , ) , , , ) , , ,
2. = 3 + 4 = 25 = 5cm = + = 12 + 5 = 144 + 5 = 169 = 13 cm 3. = 12 + (17 12) = 144 + 5 = 144 + 25 = 169 = 13 cm 4. ) = + = 12 + 9 = 144 +81 = 225 = 15cm ) = + = 8 + 15 = 64 + 225 = 289 = 17 cm ) = = 0,533,
5. = + = 6 + 4 = 52 = = + = 52 + 25 =77 = AE = + BE = 6 + 5 = 61 = 6. = + = 18 + 14 = 520 22,8cm. = 0,438, =24
387
~
4.2 1) ) : 5 2) ) : 234 cm 3) ) : 628 cm ) : 6 ) :314 cm ) : 1256 cm ) : 9 t t t t t
t t t t t
1. = + 2 = (2 80 +2 40) 30 + 2 80 40 = 13.600cm 2. ) = + = (2 25 + 2 15) 2,5 + 25 15 = 575m ) = 25 = 625 cm = 0,0625 m 575 : 0,0655 = 9.200 9.200 0,30 = 2760 3. = ( ) () 192 = 3 8 = =8cm
. . = + 2 = 192 + 2 27,71 = 247,42cm 4. = = + 2 = (0,8 + 0,6 + 0,45 + 0,45 + 0,6) 1,2 + 2 0,56 = 4,6m 4,6 15 = 69. 5. = 2 3,14 8 16 + 2 3,14 8 = 1205,76cm 6. = 50,24 20 = 1004,8 cm L = 2 50,24 = 2 3,14 = 8cm = + 2 = 1004,8 + 2 3,14 8 = 1406,72 cm
388
7. = 2 3,14 0,10 1,50 = 0,942 m 50 0,942 = 47,1 m 3 47,1 = 141,3 . 8. ) = 2 3,14 4 3 = 75,36 m = 75,16 + 2 3,14 4 = 175,64 m ) = 6cm = 452,16 cm = 678,24 cm ) = 5 cm = 314 cm = 471 cm 9. = + 2 = 2 3,14 0,4 1,2 + 2 3,14 0,4 = 4,0192m 1000 1000 4,0192 = 4019,2m 10% 4019,2 : 0,9 = 4465,77m 1,5 4.465,77 = 6698,65 . 10. ) = 2 3,14 0,15 0,6 = 0,5652m . ) = 10 18 = 180 m 180 : 0,5652 = 319 .
4
4.3 1) 2) 3) ) , ) , 25 cm2 8 200 cm3 ) 49 cm2 6cm 294 cm3 78,5cm2 4cm 314cm3 25cm2 4cm 100cm2 16cm2 9cm 144cm3 32dm 50cm 160dm3t t t t t
389
~
t t t t t
1. ) , : V = V = 5 V = 53 135 = 5 3 3 = 27 3 = 33 = 3cm. ) = + 2 = 4 5 + 2 2 = 20 32 + 2 32 = 198cm2 2. ) = + = 12 +16 = 400 =20 cm ) = + 2 = 4
) V = () () =
20 = 1.920cm3
3. ) .. : = 24 + 10 = 676 = 26cm. = = 26 cm. ) = (2 0,26 + 0,3) 2 + 2 ) V = 2 = 0,192m3
= 1,832m
4. ) .. . : = = 5 4 = 9 = 3cm. = 2 3 = 6cm = + 2 = (5 + 5 + 6) 15 + 2 ) V = 15 = 180cm3
5cm 4cm
5cm
= 264cm
5. ) V = 3,14 0,32 0,8 = 0,22608m3 = 226,08dm3 ) = 150mm = 0,15m V = 3,14 0,15 0,4 = 0,02826m3 = 28,26dm3 6. = 2 452,16 = 2 3,14 24 = = 3cm.
390
V = = 3,14 3 24 = 678,24cm3.
7. V = 3,14 3,5 10 = 384,65cm3 V = 2 384,65 = 769,3cm3 8. = 1 = 3,14 10 = 314 cm2 = 2 314 = 2 3,14 = 100 = 10cm. V = 3,14 10 5 = 1570 cm3 9. V = V V = 3,14 1,75 8 3,14 1,5 8 = 20,41m3
4
4.4 1) ) , ) , ) , ) , ) , ) t t t t t
1. V = 15 20 = 1500cm3
t t t t t
2. =
( ) ()
=
5 12 10 = 300cm
3. ) =
(4 10) 12 = 240 cm
) = + = 240 + 10 = 340 cm ) .. . : = +
391
~
12 = + 5 = 144 25 = 119 = V = 10 10,9 = 545cm3
10cm 10cm
4. . . 9,16cm 8cm 10cm 160cm2 195,41cm3 8cm 12cm 10cm 240cm2 384cm3 10cm 13,26cm 12cm 318,24cm2 1758,276cm3
5. .. () () 17 = + 8 = 289 64 = 225 = 15cm 15 = + 8 = 225 64 = 161 V = 16 12,68 = 1623,04cm3
6. ) .. : 5 = + 2,5 = 18,75 =
5cm 5cm
5cm
= ) V =
(3 5 ) 4,33 = 32,475cm
2
(. ) () =
4,1 = 14,79cm3
7.
392
V2 = 4 V1
8. ) = 10 4 = 84 = ) = (6 8) 9,16 = 219,84cm2
9,16cm
4
) .. , . = 8 4 = 64 16 = 48 = = 166,08cm2
= + = 219,84 + 166,08 = 385,92cm2 ) = 10 8 = 36 = 6cm V = 166,08 6 = 332,16cm3.
9. .. = 8 + 6 = 100 = 10cm () = 5 12 + (4 12) 10 = 960cm2
V = V V = 123 10. ) )
12 8 = 1344cm3
4.5 1) ) , ) , ) , ) , ) , ) ,t t t t t
) , )
t t t t t
1. = + ( = + = 8 + 6 = 100 = 10cm) = 3,14 6 10 + 3,14 6 = 301,44cm2
393
~
2. = 226,08 = 3,14 9 = 8cm = = 3,14 8 = 200,96cm2 3. 30 = = 6cm
30 =
= + = 3,14 3 6 + 3,14 3 = 84,78cm2. V = = 3,14 3 5,19 = 48,89cm3
4. ) L= 2 18,84 = 2 3,14 = 3dm. = 3,14 3 5 + 3,14 3 = 75,36 cm2 ) = + 5 = + 3 = 25 9 = 16 = 4dm V= = 3,14 3 4 = 37,68dm3
5. ) = + =50 + 10 = 2500 + 100 = 2.600 = .. = = 3,14 10 51 = 1601,4 cm2 .. = 2 = 2 3,14 12,5 5 = 392,5cm2 = 1601,4 + 392,5 + (3,14 12,5 3,14 10) = = 1993,9 + 3,14 (156,25 100) = 1993,9 + 176,625= = 2170,525cm2 ) V = = 3,14 10 50 = 5233,33cm3
V = = 3,14 12,5 5 = 2453,125cm3 V = 5233,33 + 2453,125 = 7686,455cm3. 6. ) V = = 3,14 10 10 =1046,66cm3
V = = 3,14 10 12 = 3768cm3 V = 1046,66 + 3768 = 4814,66 cm3. ) .. = 2 = 2 3,14 10 12 = 753,6 cm2 = + = 10 + 10 = 200
394
=
14,1cm
.. = = 3,14 10 14,1 = 442,74cm2 = 753,6 + 442,74 = 1196,34 cm2 50 50 1196,34 = 59817 cm2 = = 5,9817 m 30 5,9817 = 179,451 . 7. 19,625 = 3,14 3 = 6,25 = 2,5m = ( = + = 3 + 2,52 = 15,25 = 3,9m) = 3,14 2,5 3,9 = 30,615 m . 8. = =9,
4
= 9 29 20 = 180 9 = 180 = 20cm = 29cm = + 29 = + 20 = 841 400 = 441 = 21cm V = = 3,14 20 21 = 8792cm3 = 8,792dm3 ()
9. .. : =20 + 20 = 800 = = = 14,14cm
= 2 E = 2 = 2 3,14 14,14 20 = 1775,984cm V = 2 V = 2 V = V1 V2 = = 2 3,14 14,14 14,14 = 5918,15cm3
3,14 6 15
3,14 6 10 = 188,4cm3.
395
~
4.6 1) , 2) , 3) , 4) , 5) , 6) t t t t t
1. = 4 = 4 3,14 8 = 803,84cm V = 2. ) L = 2 50,24 = 2 3,14 ) = 803,84cm ) V = 2143,57cm3 3.
t t t t t
= 8cm
4. V = 113,04 = 3,14 3
:
3 = 27
3 = 33
= 3cm
= 4 = 4 3,14 3 = 113,04cm 5. = 4 = 4 3,14 9 = 1017,36m 5 1017,36 = 5086,8 . 6. = 1256 = 3,14 = 400 = 20cm.
396
V =
3 =
3,14 203 33493,33cm3.
4
7. = 4 = 4 3,14 6 = 452,16cm2. 8. V = 3 = 3,14 153 = 14130cm3 = 14,13 .
9. ) =
+ = 3 = 3 3,14 25 = 5887,5cm
) V =
3,14 253 = 32708,33cm3
10. ) V =
3 =
3,14 153 = 14130cm3
) V = 3 = 303 = 27000cm3 ) V = 27000 14130 = 12870 cm3 11. 1 , .. 8 =1 + 4 1 = 64 16 1 = 48 1 = E = 1 = 3,14 48 = 150,72cm.
1 1 ), ), ), () 2 ) = 4 + 3 = 25 = 5cm ) x =
t t t t t
3 ) = + = 6 + 8 = 100 = 10cm.
397
~
= E + 2 = (6 + 8 + 10) 15 + 2 ) V = 4 ) E = (4 16) 15 = 480cm 15 = 360cm3
= 408cm
) .. : 15 = + 8 = 15 8 = 225 64 = 161 = 12,68cm V = 16 12,68 = 3246cm
2 1 ), ), ), ) ()
t t t t t
2 V = = 3,14 0,4 7 = 3,5168m3 150 3,5168 = 527,52 . 3 ) E = 502,4 = 3,14 20 = 8 cm ) = + 20 = + 8 = 400 64 = 336 = 18,33cm ) V = = 3,14 8 18,33 = 1227,86cm3
4 V = V V = 203
3,14 103 = 8000 4186,66 = 3813,34cm3.
398
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4.3 O
1.
429
O OO O
2.
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200 0 200 1 200 2 200 3 200 4
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437
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1. 2.
1. 2. ) 12 m2 = 120.000 cm2, ) 175 dm2 = 17.500 cm2, ) 456 m2 = 4.560.000 cm2,) 136 m2 = 1.360.000 cm2, ) 3 km2 = 30.000.000.000 cm2, ) 1750 m2 = 17,5 cm2, ) 256 km2 = 2.560.000.000.000 cm2 ) 17 dm2 = 170.000 mm2, ) 236 cm2 = 23.600 mm2
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5.
6. 1.3 O
1.
438
O 1
1. 2. 3. 4.x x x
x
x
5.
6. 7.
8.
439
O OO O
9.
10.
11. 12.
13.
14.
15. 16.
440
O 1
17.
400 cm2. 200 cm2 100 cm2.
1.4 O
1.
2.
3.
4.
1. 2.
441
O OO O
3.
4.
5. 6. 7.
8.
442
O 1
9.
443
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O 2o2.1 O
1.
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3.
444
O 2
4.
5.
6.
7.
445
O OO O
2.2 O
1. 6. 7.
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3.
4.
5.
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446
O 2
2.3 , O
1.
2.
1.
2.
447
O OO O
3.
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5.
6.
7.
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448
O 2
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2.
3.51
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O OO O
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54
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2.6 O
1. 3. 1. 4.)
2.
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+ )
+
+ +
2. 3. 4. 5. 6.F2 + F 3 F + 1 F + 2 F3
F3
F1
F2
55
453
O OO O
7.
A
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