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Ɂɚɞɚɧɢɟ 5. Ɉɩɪɟɞɟɥɢɬɶ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ, ɫɨ-
ɫɬɚɜɥɟɧɧɨɝɨ ɢɡ ɩɪɨɤɚɬɧɵɯ ɩɪɨɮɢɥɟɣ. ɇɟɨɛɯɨɞɢɦɵɟ ɞɚɧɧɵɟ ɜɡɹɬɶ ɢɡ
ɬɚɛɥ. 5.
ȼɚɪɢɚɧɬɵ ɤ ɡɚɞɚɧɢɸ 5
1
Y
ɚ
2
Y
ɏ
ɏ
3
Y
4
ɚ
Y
c
c
ɏ
b
b
ɏ
ɚ
5
Y
6
Y
ɚ
a
ɏ
ɚ
b
41
ɏ
ȼɚɪɢɚɧɬɵ ɤ ɡɚɞɚɧɢɸ 5
Y
Y
7
Y
8
Y
b
ɚ
ɏ
a
9
Y
10
c
Y
ɏ
ɚ
b
c
b
ɏ
ɏ
11
ɚ
ɏ
12
b
ɏ
a
42
c
ȼɚɪɢɚɧɬɵ ɤ ɡɚɞɚɧɢɸ 5
ɭ
7
–
– –
–
– –
13
Y
14
Y
ɏ
15
Y
ɏ
a
16
a
Y
ɏ
ɏ
a
ʋ
ɜɚɪɢɚɧɬɚ ʋ ɫɯɟɦɵ ʋ ɲɜɟɥɥɟɪɚ
1 2 3 4 5 6
1
2 12 8/5 6
3
4 18 7,5/5 24
5
6 14 7/4,5 10 64 60
7
8 14 8/5 20 140 60
9 5 30 11/7 18 200 –
1
2
3
4
8 7/7,5 10 –
14 7/4,5 20 –
8 4/2,5 14 50 40
10 7/4,5 12 120 40
ʋ
ɝɨɥɤɚ ɚ (ɦɦ) b (ɦɦ) ɫ (ɦɦ)
b
Ɍɚɛɥɢɰɚ 5
43
Ɉɤɨɧɱɚɧɢɟ ɬɚɛɥɢɰɵ 5
7
–
– –
–
– –
–
–
–
– –
–
– –
–
– –
1 2 3 4 5 6
10 5 8 6,3/4 12 120 –
11
12 16 5,6/3,6 12
13
14 8 4/2,5 8
15
16 12 5,6/3,6 22 64 80
17
18 10 4,5/2,8 14 54 46
19
20 18 14/9
21
22 16 10/6,3 16
23
24 18 4,5/2,8 24 200 100
25
26 20 11/7 26
27
28 30 16/10 30
29
30 10 4/2,5 20
6
7
8
9
10
11
12
13
14
15
10 3,2/2 10 –
12 3,2/2 20 –
16 6,3/4 24 70 100
18 11/7 20 80 110
8 7/4,5 – 100 –
200 –
10 7/4,5 12 –
10 4/2,5 20 100 50
12 7/4,5 18 –
10 7/4,5 22 –
14 8/5 25 –
ɉɪɢɦɟɪ ʋ 4
Ɉɩɪɟɞɟɥɟɧɢɟ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɩɥɨɫɤɨɝɨ ɫɟɱɟɧɢɹ.
ɇɚɣɬɢ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɩɥɨɫɤɨɝɨ ɫɥɨɠɧɨɝɨ ɫɟɱɟɧɢɹ,
ɟɫɥɢ
ɚ = 3 ɫɦ; b = 4,8 ɫɦ; d = 6 ɫɦ; r = 2 ɫɦ.
44
Ɋȿɒȿɇɂȿ
1. Ɋɚɡɨɛɶɟɦ ɫɥɨɠɧɨɟ ɫɟɱɟɧɢɟ ɧɚ ɱɟɬɵɪɟ ɩɪɨɫɬɵɟ: 1 – ɩɪɹɦɨɭɝɨɥɶ-
ɧɢɤ, 2 – ɤɜɚɞɪɚɬ, 3 – ɬɪɟɭɝɨɥɶɧɢɤ, 4 – ɤɪɭɝɥɨɟ ɨɬɜɟɪɫɬɢɟ (ɪɢɫ. 19).
2. Ɉɩɪɟɞɟɥɹɟɦ ɩɥɨɳɚɞɢ ɩɪɨɫɬɵɯ ɱɚɫɬɟɣ ɫɟɱɟɧɢɹ.
4,8 6
bd
443672;Aad ɫɦ=⋅=⋅⋅=
1
22 2
2
636 ;Ad ɫɦ===
2
⋅⋅
A ɫɦ
== =
3
22
22 2
π
=⋅ = ⋅ =
4
3,14 2 12, 56 .Ar ɫɦ
14, 4 ;
3. Ɉɩɪɟɞɟɥɹɟɦ ɩɥɨɳɚɞɶ ɜɫɟɝɨ ɫɟɱɟɧɢɹ:
= Ⱥ1 + Ⱥ
Ⱥ
+ Ⱥ3 – Ⱥ
2
= 72 +36 + 14,4 – 12,56 = 109,84 ɫɦ
4
2
.
4. Ɉɩɪɟɞɟɥɹɟɦ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɨɜ ɬɹɠɟɫɬɢ ɫ
, ɫ2, ɫ3, ɫ
1
ɩɪɨɫɬɵɯ
4
ɫɟɱɟɧɢɣ ɢ ɢɯ ɩɥɨɳɚɞɢ:
6
d
x ɫɦ===−
1
22
443
ɚ
ɭɫɦ
== =
1
xd ɫɦ=+=+ =
3
yd ɫɦ=− =−=
3
⋅
22
b
33
4,8
67,6;
d
64;
33
3;
6;
6
x ɫɦ===
2
y ɫɦ===
2
()(63)3;xda ɫɦ=− − =− − =−
4
44339;yaa ɫɦ=−=⋅−=
4
6
d
22
6
d
22
3;
3;
1. Ɉɩɪɟɞɟɥɹɟɦ ɫɬɚɬɢɱɟɫɤɢɟ ɦɨɦɟɧɬɵ ɩɪɨɫɬɵɯ ɱɚɫɬɟɣ ɨɬɧɨɫɢɬɟɥɶ-
ɧɨ ɨɫɢ ɯ:
S
= Ⱥɤ ɭɤ;
ɤx
2
S
= 72 ⋅ 8 = 576 ɫɦ3; S
1x
S
= 14,4 ⋅ 4 = 57,6 ɫɦ3; S
3x
= 36 ⋅ 3 =108 ɫɦ3;
2x
= 12,56 ⋅ 9 =113,04 ɫɦ3.
4x
2. Ɉɩɪɟɞɟɥɹɟɦ ɫɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɜɫɟɝɨ ɫɟɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɨɫɢ ɯ:
S
= S1x.+ S2x + S3x – S
x
= 576 + 108 + 57,6 – 113,04 = 484,56 ɫɦ3.
4x
45
3. Ɉɩɪɟɞɟɥɹɟɦ ɫɬɚɬɢɱɟɫɤɢɟ ɦɨɦɟɧɬɵ ɩɪɨɫɬɵɯ ɱɚɫɬɟɣ ɨɬɧɨɫɢɬɟɥɶ-
A
ɧɨ ɨɫɢ ɭ:
S
= Ⱥɤ ɯɤ;
ɤɭ
S
= 72 ⋅ (–3) = –216 ɫɦ3; S
1ɭ
= 36 ⋅ 3 =108 ɫɦ3;
2ɭ
= 14,4 ⋅ 7,6 = 109,44 ɫɦ3; S
S
3ɭ
= 12,56 ⋅ (–3) = –37,68 ɫɦ3.
4ɭ
4. Ɉɩɪɟɞɟɥɹɟɦ ɫɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɜɫɟɝɨ ɫɟɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɨɫɢ ɭ:
S
= S1ɭ. + S2x + S3ɭ – S
ɭ
= 216 +108 + 109,44 – (–37,68) = 39,12 ɫɦ3.
4ɭ
5. Ɉɩɪɟɞɟɥɹɟɦ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɩɥɨɫɤɨɝɨ ɫɟɱɟɧɢɹ ɫɨɝɥɚɫɧɨ ɮɨɪɦɭ-
ɥɚɦ:
S
y
x= ; y= ; x= =0,36ɫɦ; ɭ ==4,4ɫɦ
ccc c
A A 109,84 109,84
S
x
39,12 484,56
.
6. Ɉɛɨɡɧɚɱɢɦ ɰɟɧɬɪ ɬɹɠɟɫɬɢ – ɬɨɱɤɭ ɋ (0,36; 4,4) ɫɦ ɧɚ ɫɟɱɟɧɢɢ
(ɪɢɫ. 19).
7. Ⱦɚɧɧɵɟ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɜɫɟɝɨ ɫɟɱɟ-
ɧɢɹ ɡɚɩɢɫɵɜɚɟɦ ɜ ɬɚɛɥ. 6.
Ɍɚɛɥɢɰɚ 6
ɉɪɨɫɬɵɟ ɫɟɱɟɧɢɹ Ⱥ
ɤ
ɯɤ ɭɤ
Ⱥ
⋅ ɯɤ Ⱥɤ ⋅ ɭɤ
ɤ
ɉɪɹɦɨɭɝɨɥɶɧɢɤ 72 –3 6 –216 432
Ʉɜɚɞɪɚɬ 36 3 3 108 108
Ɍɪɟɭɝɨɥɶɧɢɤ 14,4 7,6 4 109,44 57,6
Ʉɪɭɝ 12,56 –3 9 –67,68 113,04
ȼɫɟ ɫɟɱɟɧɢɟ 109,84 0,36 4,41 39,12 484,56
ɉɪɢɦɟɱɚɧɢɟ: ɨɩɪɟɞɟɥɟɧɢɟ ɤɨɨɪɞɢɧɚɬ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɜɫɟɝɨ ɫɟɱɟɧɢɹ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ, ɢɫɩɨɥɶɡɭɹ ɫɥɟɞɭɸɳɢɟ ɮɨɪɦɭɥɵ:
xAx AxAx
++−
11 22 33 44
x
=
c
AA A A
++−
1234
;
46
A
yAy AyAy
y
++−
11 22 33 44
y
=
c
AA A A
++−
1234
.
a
a
4a
R
C
4
C
1
C
C
C
2
3
d
0
d
d
b
x
Ɋɢɫ. 19.
ɉɨɞɫɬɚɜɢɦ ɜ ɷɬɢ ɮɨɪɦɭɥɵ ɞɚɧɧɵɟ ɢɡ ɬɚɛɥ. 7 ɢ ɨɩɪɟɞɟɥɢɦ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɫɥɨɠɧɨɝɨ ɫɟɱɟɧɢɹ:
–
216 +108 +109,44 +37,68 39,12
x = = = 0,36 ɫɦ;
c
72+ 36 +14,4 12,56 109,84
–
432+108 + 57,6 113,04 484,56
y= = =4,41ɫɦ.
c
72+ 36 +14,4 12,56 109,84
–
–
Ɉɬɜɟɬ: ɯ
= 0,36 ɫɦ; ɭɫ = 4,41 ɫɦ.
ɫ
47
3,2
3
ɉɪɢɦɟɪ ʋ 5
7
A
A
ɭ
z
ɯ
Ɉɩɪɟɞɟɥɟɧɢɟ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ,
ɫɨɫɬɚɜɥɟɧɧɨɝɨ ɢɡ ɩɪɨɮɢɥɟɣ ɩɪɨɤɚɬɚ.
Ɉɩɪɟɞɟɥɢɬɶ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ, ɫɨɫɬɚɜɥɟɧɧɨɝɨ ɢɡ ɲɜɟɥɥɟɪɚ
ʋ 12, ɧɟɪɚɜɧɨɛɨɤɨɝɨ ɭɝɨɥɤɚ ʋ 10/6,3 ɢ ɩɨɥɨɫɵ ɲɢɪɢɧɨɣ ɚ = 1 ɫɦ.
Ɋȿɒȿɇɂȿ
ɂɡ ɬɚɛɥɢɰ ɫɨɪɬɚɦɟɧɬɨɜ ɩɪɨɤɚɬɧɵɯ ɩɪɨɮɢɥɟɣ ɛɟɪɟɦ ɜɫɟ ɧɟɨɛɯɨɞɢɦɵɟ ɞɚɧɧɵɟ ɞɥɹ ɩɪɨɮɢɥɟɣ ɩɪɨɤɚɬɚ ɢ ɱɟɪɬɢɦ ɜ ɦɚɫɲɬɚɛɟ ɫɯɟɦɭ ɫɟɱɟɧɢɹ
ɫ ɭɤɚɡɚɧɢɟɦ ɨɫɧɨɜɧɵɯ ɪɚɡɦɟɪɨɜ (ɪɢɫ
y
C
2
0
1,42
1,34
C
C
1
12
Ɋɢɫ. 20.
ɋ
ɤɚɠɞɨɝɨ ɩɪɨɫɬɨɝɨ ɫɟɱɟɧɢɹ. ɋɨɫɬɚɜɥɹɟɦ ɫɥɟɞɭɸɳɭɸ ɬɚɛɥ. 7.
3
ɇɚɢɦɟɧɨɜɚɧɢɟ ɩɪɨɮɢɥɹ Ⱥ
(ɫɦ2) ɯɤ (ɫɦ) ɭɤ (ɫɦ)
ɤ
ɒɜɟɥɥɟɪ ʋ 12 13,3 6 –1,54 79,8 –20,482
ɍɝɨɥɨɤ ʋ 10/6,3 9,59 8,77 1,42 84,104 13,618
ɉɨɥɨɫɚ 1×11,5
11,5 12,5 0,55 143,75 6,325
ɐɟɧɬɪ ɬɹɠɟɫɬɢ ɲɜɟɥɥɟɪɚ ɋ
ɥɟɪɚ, ɩɨɷɬɨɦɭ:
h
x= =6ɫɦ,
1
2
. 20).
1
Į
t =
,8 ɦɦ ȼ = 100 ɦɦ
d = 4,8 ɦɦ ɚ = 10 ɦɦ
h = 120 ɦɦ b = 63 ɦɦ
6,5
b = 52 ɦɦ
C
x
3
5,2
= 13,3 cɦ2
= 1,54 cɦ
0
= 9,59 cɦ2
3,23
0 =
= 1,42 cɦ
0
1. Ɋɚɡɛɢɜɚɟɦ ɫɨɫɬɚɜɧɨɟ
ɫɟɱɟɧɢɟ ɧɚ ɩɪɨɫɬɵɟ ɫɟɱɟɧɢɹ –
ɲɜɟɥɥɟɪ (ɪɢɫ. 21, ɚ), ɭɝɨɥɨɤ
(ɪɢɫ. 21, ɛ), ɩɨɥɨɫɭ ɢ ɨɩɪɟɞɟɥɹɟɦ ɰɟɧɬɪɵ ɬɹɠɟɫɬɢ ɋ
Ɍɚɛɥɢɰɚ 7
Ⱥ
⋅ ɯɤ Ⱥɤ ⋅ ɭɤ
ɤ
ɧɚɯɨɞɢɬɫɹ ɧɚ ɫɟɪɟɞɢɧɟ ɜɵɫɨɬɵ ɲɜɟɥ-
1
ɭ1 = –z0 = –1,54 ɫɦ.
, ɋ2,
1
48
ɐɟɧɬɪ ɬɹɠɟɫɬɢ ɧɟɪɚɜɧɨɛɨɤɨɝɨ ɭɝɨɥɤɚ ɋ
ɟɦ ɯ
= 1,42 ɫɦ, ɭ0 = 3,23 ɫɦ, ɜɡɹɬɵɦɢ ɢɡ ɬɚɛɥɢɰɵ (ɉɪɢɥɨɠɟɧɢɟ 2 ɫɬɪ.
0
68). ɉɪɢɱɟɦ ɪɚɫɫɬɨɹɧɢɟ ɯ
ɭɝɨɥɤɚ, ɚ ɪɚɫɫɬɨɹɧɢɟ ɭ
ɜɫɟɝɞɚ ɨɬɤɥɚɞɵɜɚɟɬɫɹ ɜɞɨɥɶ ɦɟɧɶɲɟɣ ɩɨɥɤɢ
0
– ɜɞɨɥɶ ɛɨɥɶɲɟɣ ɩɨɥɤɢ ɭɝɨɥɤɚ.
0
ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɚɫɫɬɨɹɧɢ-
2
ɉɨɷɬɨɦɭ:
ɯ
= h – y0 = 12 – 3,23 = 8,77 ɫɦ,
0
y
= ɯ0 = 1,42 ɫɦ.
0
ɐɟɧɬɪ ɬɹɠɟɫɬɢ ɋ
ɩɨɥɨɫɵ ɧɚɯɨɞɢɬɫɹ ɧɚ ɩɟɪɟɫɟɱɟɧɢɢ ɞɢɚɝɨɧɚɥɟɣ
3
ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ:
x = h+ = 12+0,5 = 12,5 ɫɦ,
3
y = b = 6,3 = 0,55 ɫɦ.
32
a
2
b+b
12
––
22
5,2 +6,3
z
0
x
0
C
h
B
C
b
b
ɚ) ɛ)
Ɋɢɫ. 21.
2. Ɉɩɪɟɞɟɥɹɟɦ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɜɫɟɝɨ ɫɟɱɟɧɢɹ ɫɨɝɥɚɫɧɨ ɮɨɪɦɭɥɚɦ:
xAxAxA
++
332211
x
=
c
++
;
AAA
321
49
0
y
yAyAyA
++
332211
y
=
c
++
.
AAA
321
ɉɨɞɫɬɚɜɥɹɟɦ ɜ ɷɬɢ ɮɨɪɦɭɥɵ ɞɚɧɧɵɟ ɢɡ ɩɨɫɥɟɞɧɢɯ ɞɜɭɯ ɫɬɨɥɛɰɨɜ
ɬɚɛɥɢɰɵ ɢ ɨɩɪɟɞɟɥɹɟɦ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɫɨɫɬɚɜɧɨɝɨ ɫɟɱɟɧɢɹ:
75,143104,848,79
x
=
c
++
5,1159,93,13
++
654,307
==
39,34
;ɫɦ946,8
325,6618,1382,204
y
=
c
++−
5,115,93,13
−+
539,0
−
=
−=
39,34
.ɫɦ016,0
ɉɨ ɷɬɢɦ ɤɨɨɪɞɢɧɚɬɚɦ ɫɬɪɨɢɦ ɰɟɧɬɪ ɬɹɠɟɫɬɢ – ɬɨɱɤɭ ɋ(8,946;
–0,016) ɫɦ ɫɨɫɬɚɜɧɨɝɨ ɫɟɱɟɧɢɹ.
Ɉɬɜɟɬ: ɯ
= 8,946 ɫɦ; ɭɫ = –0,016 ɫɦ.
ɫ
2. ɄɈɇɌɊɈɅɖɇɕȿ ȼɈɉɊɈɋɕ
1.
ɑɬɨ ɬɚɤɨɟ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɬɟɥɚ?
2.
Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɬɪɟɭɝɨɥɶ-
ɧɢɤɚ?
3.
ɑɬɨ ɬɚɤɨɟ ɫɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ? ȿɞɢɧɢɰɵ ɢɡɦɟɪɟɧɢɹ.
4.
Ʉɚɤ ɨɩɪɟɞɟɥɢɬɶ ɫɬɚɬɢɱɟɫɤɢɣ ɦɨɦɟɧɬ ɫɟɱɟɧɢɹ ɫɥɨɠɧɨɣ ɮɨɪɦɵ?
5.
Ɏɨɪɦɭɥɵ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɫɟɱɟɧɢɹ.
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