TALLER Nº4 VIAS
PRESENTADO A: ING. DANIEL CONTRERAS BARRETO
PRESNETADO POR: LUZ ELENA QUINTERO 1111382 JUAN DAVID CEPEDA CEPED A 1111 1111370 370 ALEJANDRA ALEJANDRA ARCINIEGA ARCINIEGAS S 111 111136 136 DIEGO ALEJANDRO DIAZ 1111333
INGENIERIA CIVIL
UNIVERSIDAD !RANCISCO DE PAULA SANTANDER SANTANDER CUCUTA" NORTE DE SANTANDER SEPTIE#BRE DE 201$
PROBLEMAS PROPUESTOS DE LA JAMES CARDENAS GRISALES
PROBLEMA 3.8 Datos: P%&% '% !()*&% 3.86" +, -(,, '% +()*(,-, (/&%(: C&,%%+ ,' POT 1 C&,%%+ ,' P' 1 C&,%%+ ,' P' 2 C&,%%+ ,' POT 2 A+(+% ,' POT1 D(+-%(% P' 1.P'91 D(+-%(% P' 2.P'92 C*,&%+
Ca"%u"ar: L% ,*%( , ,%',.
So"u%i&'
5 N: 378.180" E: 246.860 5 N: 23.40" E: 184.070 5 N: 1$3.10" E: 461.620 5 N: 24$.120" E: $72.370 5 487.820 513.100 5 3$.600 5 5 10
Figura 3.8 Pro!"#$a 3.8
Pl 1 . Pl 2
( 86,03) 2 + ( 277,55) 2 5 20"$77
5 5
. POT 1 Pl 2
C+ A 5
a
(138,24) 2 + ( 62,79) 2 5 1$1"832
5
POT 1. Pl 1
2
( 224,27) 2 + ( 214,76) 2 5 310"$14
+ c2 − b2 2ac
A 5 82;3697299 82;3697299
<1 5 180; = C <1 5 180; = 82;47936.7299 5 7;12923"2899
Cur(a No. ) T2 5 R2 -%)
∆ ∆ 2 2
T 1
R1 5
100
∆ ∆ tan 2
G1 5 2 %&+,
L 5
ο R1 5 tan 97 12'23.28' ' 2
C 1
G1 5 6;20911.3399 6;20911.3399
2 R1
(10) ( 97ο 12'23.28' ')
L2 5 14"476
ο
6 30'11.33' '
A+(+% ,' PT1 5 A+ POT 1
. POT 1 PC 1
L1
A+(+% ,' PT1 5 4 87"820 $1"832 131"73 A+(+% ,' PT1 5 $ 081"128
Cur(a No. * Pl 2 . POT 2
5
2 ( 91,21) 2 + (110 110,75) 5 143"474
R1 5 88"1$2
Pl 1. POT 2
( 5,18) 2 + ( 388,3) 2 5 388"33$
5
5 2 2 > 2 ?@ ?@ C+ N
+ 143,4742 − 388,3352 2( 290,577) (143,474) 2
C+ N 5
290,577
N 5 123;1891.299 <2 5 180; = 123;1891"2 5 $6;41940.899 <2 5 $6;41940.899 T2 5 R2 -%)
∆ ∆ 2 2
T 2
R2 5
70
∆ ∆ tan 2
G 5 2 %&+,
ο R2 5 tan 56 41'40.8' ' 2
10 2 x129.747
G 5 4;2$91499
L 5 128"3$4 A+(+% ,' PT2 5 A+ PT2 ( Pl 1. Pl 2
− T 1 − T 2 ) L
A+(+% ,' PT2 5 $ 081"128 120"$77 128"3$4 A+(+% ,' PT2 5 $ 330"0$ P%&% '% % 2 2 5 20"33$ 2 3$"60 2 > 2 ?20"33$@?3$"60@ C+ $6;41940.899 5 272"417 ο
Sen56 41'40.8' '
272,417
5
Senβ
35,60
β 5 6;16913.0799
R2 5 12"747
L2 5 13"102 272"417 2 > 2 ?13"100@?272"417@ C+ 0;$691"2199 5 308"3 ο
Sen90 56'10.36' '
307,9
5
Sen( m )
5 62;10913.399
272,417
<91 5 180; = 62;10913"399 <2 5 117;4946.699 R91 5 66"366 G 5 2 %&+,
10
G 5 8;3892.$799
2( 66,36)
L 5 136"3$3 A+(+% ,' PT91 5 A+ PC 1 T1
Pl 1. PC 1
L91
A+(+% ,' PT91 5 4 31"6$2 100 2"100 136"3$3 A+(+% ,' PT91 5 $ 17"10$ T2 5 R92 -%)
∆'2 2
T 2
R92 5
70
∆ 2
R92 5 tan 36 6'30.08' ' 2 ο
tan
G 5 2 %&+,
10
(
ο
2 36 6'30.08' '
)
G 5 1$;$696.899
L92 5 22"636 A+(+% ,' PT92 5 A+ PT9 1
PT '1.PC '2
L92
PT '2 .PT 2
A+(+% ,' PT92 5 $ 302"846 127" 22"68$ 3$"6 A+(+% ,' PT92 5 $ 383"688
PROBLEMA 3.+
R92 5 214"81
Datos: P%&% '% !()*&% 3.87" +, -(,, '% +()*(,-, (/&%( %((%': C&,%%+ , B 5 N: 421.360" E: 376.840 C&,%%+ , C 5 N: 62.880" E: $34.60 A(*- , AB 5 334;93899 A(*- , CD 5 8;$094299 D(+-%(% AB 5 101 D(+-%(% CD 5 126
Ca"%u"ar: L% ,*%( , ,%', ,' E, 2 , ,' E, 1.
Figura 3.8, Pro!"#$a 3.+ So"u%i&' C α 5 2$;$092299 <1 5 α + θ <1 5 2$;$092299 37;10922.$99 <1 5 63"012362 <1 5 63;00944.$99
BE EC
-% θ 5
158,120
θ 5 37"17217
208,520
C%'*' TB 5 R B -%
∆1 2
RB 5 110
TB 5 67"424
P CB A 5 AB − T B P A 5 CB
101 > 67"424
P CB A 5
33"$76
A+ PCB 5 A+ A
P A CB
A+ PCB 5 2 83$"460 33"$76 A+ PCB 5 2 86"036
C
C%'*' G B 5 2 %&+,
2 R B
GB 5 $"210$03 GB 5 $;12937.8199
C%'*' L B 5 120"33
A+ PTB 5 A+ PCB LB A+ PTB 5 2 86"036 120"33 A+ PTB 5 2 8"6 <1 5 <2 <2 5 63;00944.$99 TB 5 T9B
T9B 5 67"424
θ 5 37;10922.$99
<1 PI. !PI9 5 S, < 1 5
24m PI 1 PI
24
PI . PI ' 5
Sen( 63ο 00'44.5' ')
A. PCB' 5
AB PI.PI9 > TB9
A. PCB' 5
101 26"700 = 67"424
A. PCB' 5
60"276
A+ PCB9 5 A+ A A
PI . PI ' 5
P CB
A+ PCB9 5 2 83$"460 60"276 A+ PCB9 5 2 8$"736
LB 5 L9B
L9B 5 120"33
A+ PT9B 5 A+ PCB L9B A+ PT9B 5 2 8$"736 120"33 A+ PT9B 5 3 016"66
θ 5 8;$094299
<1 5 0; φ − θ <1 5 0; 8;$094299 = 37;10922"$99 <1 5 61"3387$ <1 5 61;2091.$99 C%'*' T9 5 R9 -% T9 5 $"303
∆3 2
R9 5 100
26"700
< PI !P9I
PI . F
5
-% < 1 5
24m PI . F
24 PI .F
tan ( 63 00'44.5' ') ο
< BEC
5 12"222
BC 5
( BE ) 2 + ( EC ) 2
BC 5
(158,120) 2 + ( 208,520) 2
BC 5
261"62
PI '. PI '2
5
BC = PI . F = KC
PI '. PI '2
5 261"62 > 12"222 > 13"11
PI '. PI '2
5 236"3$1
PT B '.PC C
5
PT B '.PC C
5 236"3$1 > 67"424 > $"303
PT B '.PC C
5 10"626
PI .' PI '2
A+ PC9C 5 A+ PT9 B
= T9B > T9C
PT ' B .PC C
A+ PC9C 5 3 016"66 10"624 A+ PC9C 5 3 126"23
C
C%'*' G9 C 5 2 %&+, G9C 5 $"73168 S, <1 5
24 PI . PI '
2 R B
G9 C 5 $;439$$.0899
PI . PI ' 5
24
< C PI9 2
KC
-%
KC
tan ( 61 20'19.5' ') ο
S, <3 5
5
26"33
24
24
5
PI '2 C
PI . PI ' 5
Sen( 63 00'44.5' ') ο
KC
5 13"11
24 PI 'C 2
24
Sen( 61 20'19.5' ') ο
PI '2 C
5 27"3$1
C%'*' L C 5 107"012 A+ PT9C 5 A+ PC9 C L9C A+ PT9C 5 3 126"23 107"012 A+ PT9C 5 3 233"30$
C%'*'
γ 5 180; = < 3 γ 5 180; = 61;2091.$99 γ 5 118;3940.$99
C%'*' T 5 R -%
∆o 2
R9 5 186 <9 5 180; = γ = <% <9 5 180; = 118;3940.$99 = 32;
T 5 48"68
<9 5 2;2091"$99
senγ
< CF
To
∆o
Sen
5
Senγ
+ T ' o
Sen∆o b
?T T9@
Sen32ο
5
5
Sen(118ο 39'40.5' ')
?36"130 48"68@
5 $1"224 Sen∆o a
%5
%5
5
Sen∆o b
∆' o sen∆o
bsen
( 51,224) sen( 29ο 20'17.5' ') ο
sen32
% 5 47"363
TE 5 T %
TS 5 T9
TE 5 36"130 47"363
TS 5 48"68 $1"224
TE - 83/+3 $
TS - +++)3 $
PROBLEMA 3.)0 Datos: A,+ , '% (/&%( %% , '% !()*&% 3.88" +, ,:
D(+-%(% AB A+(+% , A C*,&%+
5131 5 0 846 5 5 $
Ca"%u"ar: L% ,*%( , ,%', ,' E, 2 , ,' E, 1.
Figura 3.88 Pro!"#$a 3.)0 So"u%i&' A+ A 5 0 846 C5$ R 5 3$ T 5 44 A+ PC A1 5 A+ A > T 5 0 802 G 5 8;11931.$299 L1 5 62"86
A+ PT A1 5 A+ PC A1 L1 5 864"6 A+ A9 5 A+ A 1 5 0 868"40
T 5 66"40 < 5 127;
66"4 5 R -%
127 2
R 5 33"10 G 5 8;3942.0399 L 5 73"31
A+ PT9 A1 5 A+ PC A1 L A+ PT9 A1 5 0 87$"31
AB
5 131
A+ B 5 0 $1"86
< 5 77; R 5 88 T 5 6" A+ PCB1 5 A+ PT A1 A9B9 A+ PCB1 5 0 881"87
G 5 3;1$921.1799 L 5 118"24
A+ PTB1 5 A+ PCB1 L
A!s PTB) - 1) 2 000) d 3 sen (103) d 2 sen ( 24 )
87
5
5
3 5 106"14
Sen( 53) 87
Sen( 53)
2 5 44"30
T3 5 T2 > 2 T3 5 2$"68 <3 5 $7;
5 2
2$"68 5 R -% = L 5 47"62 R 5 $1.$0 G 5 $;339$3.$899
A+ PC9B1 5 A+ PT B1 > L A+ PC9B1 5 0 $2"4 A' C 5
A!s
80"46
PT ' B1 -
1) 2 ++*+/
PROBLEMA 3.)) Datos: A,+ , '% (/&%( %% , '% !()*&% 3.8" +, ,:
C&,%%+ , A C&,%%+ , B C&,%%+ , B
5 N: 800" E: $00 5 N: 1000" E: $60 5 N: 00" E: 680
Ca"%u"ar: L% ,*%( , ,%', ,' E, 2 , ,' E, 1.
Figura 3.8+ Pro!"#$a 3.)) So"u%i&' C '%+ &,%%+ , A" B" H C +, *,, %'*'%& '+ %(*- H (+-%(%+ , AB H BC AB
5 208"806
A(*- A > B 5 16"6;
AC
5 20$"13
A(*- A > C 5 60"4$;
BC 5
1$6"20$
A(*- B > C 5 12"806;
Para "a %ur(a No. ) #4# )5 R1 5 66 A+ PC1 5 0 82"284
P%&% %''%& ,' < 1 <1 5 A( 0; = A( A > B <1 5 0; = 16"6; <1 5 73"301; T1 5 R1 -%
L1 5
∆1 2
π R∆
73,301ο T1 5 66 -% 2 π ( 66) ( 73,301
ο
L1 5
180
)
180
T1 5 4"106
L1
84"437
Para "a %ur(a No. * #4# )5 R2 5 37 P%&% %''%& ,' < 2 <2 5 A( B > C = A( A > B <2 5 12"806; = 16"6; <2 5 113"107; T2 5 R2 -%
L2 5
π R∆ 180
∆2 2
113,107ο T2 5 37 -% 2 L1 5
π ( 37) (113,107ο 180
T2 5 $6"014
L1 5 73"041
5
Para "a %ur(a No. 3 #4# )5 R3 5138 P%&% %''%& ,' < 3 <3 5 A( B > C = A( 0; <3 5 12"806; = 0; <3 5 3"806; T3 5 R3 -%
L3 5
∆3 2
39,806ο T3 5 138 -% 2
π R∆
L3 5
180
π (138) ( 39,806ο )
PT 1
− PC 2 5
PT 1
− PC 5 208"806 > 4"106 > $6"014
PT 1
− PC 2 5 103"686
AB
180
T3 5 4"63
L3 5 $"87$
= T1 > T2
2
PT 2
− PC 5 3
BC =
T2 > T3
PT 2
− PC 3 5 1$6"20$ > $6"014 > 4"63
PT 2
− PC 5 $0"228 3
A+ PT3 5 0 82"284 L 1 PT 1 − PC 2 L2
PT 2
− PC 3 L3
A+ PT3 5 0 82"284 84"437 103"686 73"041 $0"228 $"87$ A+ PT3 5 1 2"$$1
Para "a %ur(a No. / #4# )5 P%&% %''%& ,' < 4
<4 5 A( 0; = A( A > C <4 5 0; = 60"4$; <4 5 2"0$$; T4 5 T1
T4 5 4"106 T 4
49,106
∆ tan 4 2
R4 5
L4 5
R4 5
π R∆
L4 5
180
29,055ο tan 2 π (189,504) ( 29,055ο ) 180
R4 5 18"$04
L4
5
6"08
Para "a %ur(a No. 6 #4# )5 P%&% %''%& ,' < $ <$ 5 A( 0; = A( A > C <$ 5 0; = 60"4$; <$ 5 2"0$$; T$ 5 T3
T$ 5 4"63 T 5
49,963
∆ tan 5 2
R$ 5
L$ 5
R$ 5
π R∆
L$ 5
180
7"77$
PT 4
− PC 5 5
AC
= T4 > T$
29,055ο 2
tan
π (192,811) ( 29,055ο ) 180
R$ 5 12"811
L$
5
PT 4
− PC 5 20$"13 > 4"106 > 4"63
PT 4
− PC 5 5 106"844
PT 4
− PC 5 106"844
5
5
A+ PT$ 5 0 82"284 L 4
PT 4
− PC 5 L$
A+ PT3 5 0 82"284 6"08 106"844 7"77$ A+ PT3 5 1 13"001
E%ua%i&' 7# E$a"$# - 1) 2 *++66) E4# )5 - 1) 2 )+300) E4# *5 PROBLEMA 3.)* Datos: A,+ , '% (/&%( %% , '% !()*&% 3.0" +, ,: C&,%%+ , A C&,%%+ , B C&,%%+ , C C*&% , ,-& ! C*&% , ,-& G C*&% , ,-&+ I H C*&% , ,-& J
5 N: 1000.000" E: 1000.00 5 N: 1132.$10" E: 1030.$0 5 N: 1123.4$0" E: 26.0 5 T 5 37 " 5 10 5 R 5 32 " 5 $ 5 T 5 48 " 5 $ 55$
Ca"%u"ar: L%+ ,*%(,+ , ,%', ,,+%&(%+.
Figura 3.+0 Pro!"#$a 3.)* So"u%i&' E, 1 > E, 2 T
5 40
5 180; = < A
T3
5 40
R
5 34.82
5 123;$093799
LJ
5 41"407
RI
5 40"28
TG
5 60"182
L!
5 62"$12
R!
5 6"$
RG
5 32
L
5 6$"7378
TJ
5 22"$26
T2
5 27"83
LG
5 6"2$4
T1
5 23"18
LI
5 70"22
RS 5 42"36
< A 5 180 = $6;09399
< A 5 102;$9$799
5
( y − y ) 5 ( x − x ) 2
1
132,510 − 1000,000
2
1
1030,590 − 1000,000
θ 5 %&-% ?@ θ 5 77;09399
A+ PTB2 5 184"1704 A+ 5 0 0"$0
E+- , ((% K*, L 5 $0 " *,+ A+ (((%' ,+ 0 000 LN 5 LB1 > L L 5 73"348 > 80 5 23"348
ο
<3 5
Lc. N 180
π R B1
5
( 23,9348)(180) 5 1;2$92$"3899 π ( 70,6025)
U+ ,+-, %'& %&% %''%& L Q LK 5
∆3π RA1 ο
180
5
(19ο 25'25,38' ')π ( 86,6025) 5 2"3$8 ο
180
A+ P 5 A+ PT A1 > LQ A+ P 5 0 0"68 > 2"3$8 A+ P 5 0 061"331 5
1132,510 − 1000,00 1030,590 − 1000,00
5 4"33
θ 5 %&-% ?@ θ 5 77;09399
T F
∆ 5 6"$ R! 5 tan 2 L 5
∆
C
D
Gc
5 67"6
T J
∆ 5 22"$$ RJ 5 tan 2 LJ 5
∆
C
D
Gc
10 5 8"2; 2 R
G 5 2 %&+,
5 22"7$4
C 5 12"7416; 2 R
G 5 2 %&+,
C 5 8"61$; 2 R
G 5 2 %&+,
TG 5 60"182 L 5
∆
C
A
Gc
5 6"1823
T I
∆ 5 40"38 RI 5 tan 2 LJ 5
∆
C
E
Gc
5 70"2$
C 5 7"12; 2 R
G 5 2 %&+,
T H
C 5 8"22; 2 R
∆ 5 34"88 R 5 tan 2 LJ 5
∆
C
B
G 5 2 %&+,
5 6$"6
Gc
T1 5 TG > T! 5 23"182 (1000 − 1030,590) 2 + (1000 − 1123,459) 2
AB
5
AB
5 136
T2 5 136 > 23"182 > 37 > 48 5 27"813 (1000,590 − 926,990) 2 + (1132,511 − 1123,459) 2
BC 5 BC 5
104
T3 5 104 > 48 5 $6 SenA a
5
5
SenB b
(104 ) ( Sen ( 72ο 0'4' ') )
(
ο
Sen 99 59 '49 ' '
)
5 100"43
T3 5 100"43 > 48 > 12 5 40 SenA a
5
SenC c
5 14"6 P 5 136 > 14"6 5 121"302 T+ 5 121"303 > 48 > 22"$26 5 $0"78
E$a"$# No. ) E, N. 1 5 0 000 L J T+ LJ T4 E, N. 1 5 0 000 41"402 $0"78 70"2$ 40 E, N. 1 5 0 202"862
E, N. 3 5 0 000 L G T1 T2 L T3 E, N. 3 5 0 000 6"183 23"182 22"813 63"6 $6 E, N. 3 5 0 240"64$
E$a"$# No. * E, N. 2 5 0 000 L ! T4 E, N. 2 5 0 000 67"6 23"182 E, N. 2 5 0 01"142
E, N. 3 5 0 000 L G E, N. 3 5 0 000 6"183 E, N. 3 5 0 6"183
PROBLEMA 3.)3 Datos: A,+ , '% (/&%( %% , '% !()*&% 3.1" +, ,: C&,%%+ , A 5 N: 1000" E: 1000 C&,%%+ , B 5 N: $7" E: 111$ C&,%%+ , C 5 N: 1161" E: 1227 A(*- , CD 5 12$; A(*- , BE 5 46; R%(+ 5 R1 5 R91 5 0 T%),-,+ 5 T2 5 T92 5 2 C*,&%+ 5 5 10
Ca"%u"ar: L% ,*%( , ,%', , '% V% 2 , '% V% 1.
Figura 3.+) Pro!"#$a 3.)3 So"u%i&' 9a No. ) 5 10 <1 5 81;4491$.99 ο
T1 5 0 -%
81 44'15.9' ' 2 C
5 77"87
G 5 2 %&+,
5 6;22910.1299 2 R
L 5
(
ο
) 5 128"32
10 81 44'15.9' ' ο
6 22'10.12' '
<2 5 θ = A(*- CD 5 6;139$6.4699 3
T2 5 R2
∆2
R2 5 82"$
2
G2 5 2 %&+,
L2 5
(
5 6;$69$7"1399 2( 82,5) 10
ο
) 5 138"48
10 96 13'56,46' '
T% θ 5
ο
6 56'57.13' '
C op
θ 5 %&-%
A
20;$09$.1399
204 112
θ 2 5 61;139$$.299
112 204
θ 3 5 28;4694.99
θ 2 5 %&-%
θ 3 5 %&-%
A+ PC1 5
AB
= T1
A+ PC1 5 122"78 > 77"87 A+ PC1 5 4$
A+ PT1 5 A+ PC L A+ PT1 5 173"23
A+ PC2 5 A+ PT ET A+ PC2 5 173"2 62"8$ A+ PC1 5 236"08
A+ PT2 5 A+ PC2 L2
43 115
θ 1 5
A+ PT2 5 238"08 138"48 A+ PT2 5 374"$6
A+ PT92 5 374"$6 70"2$ A+ PT92 5 0 444"7
9a No. * <91 5
θ 1
44 5 64;309$.1399
∆
T1 5 R -%
2
5 $6"78
C 2( 90)
GS 5 2 %&+,
L 5
(10) ∆'1
BE 5 SenB CE
GS
5 1$3"83
Sen( 83ο 41') ( 232,72) Sen( 79)
=
Sen( 79)
CE 5 70"23
232,72
E-&,-%) 5
BE =
5 23$"17
T1 > T2
E-&,-%) 5 86"8 A+ PC 5
AB
= T91 5 66
A+ PT 5 A+ PC L 5 167"26
A+ PC2 5 A+ T1 E-&,-%) A+ PC2 5 167"26 86"8
A+ PC2 5 2$4"1$
A+ PT 5 A+ PC L A+ PT 5 0 407"8
E%ua%i&' 7# E$a"$# - 10 2 /0,+8 9ia *5 - 10 2 ///,+ 9ia )5 PROBLEMA 3.)/ Datos: L+ K*, %%&,, , '% !()*&% 3.2.
Figura 3.+* Pro!"#$a 3.)/ Ca"%u"ar: %@ L% ,*%( , ,%',. @ L% %+(+% ,' *- P
So"u%i&' T A1 5 $0 <1 5 ?A(*- $; = A(*- 3$;@
<1 5 60; <2 5 ?A(*- 21$; = A(*- $;@ <1 5 120; A+(+% O&(), 0 000 A ,' C%&&(' 5 16
Cur(a No. ) L,% A T
R A1 5
L A1 5
50
∆1 5 2
tan
π R∆ ο
180
60ο 5 86"602$ 2
tan
5 0"68
A+ PT A1 5 A+ PC A L A A+ PT A1 5 0 000 0"68 A+ PT A1 5 0 00"68
L,% B RB1 5 R A1 > A %&&(' 5 86"602$ > 16 5 70"602$ LB1 5
π R∆ ο
180
5 73"348
A+ PTB1 5 A+ PCB LB A+ PTB1 5 0 000 73"34 A+ PTB1 5 0 073"348
Cur(a No. * L,% A 152 − 50
T 2
R A2 5
L A2 5
∆ 5 tan 2 2 π R∆ ο
180
120ο 5 $8"887 tan 2
5 123"3383
A+ PT A2 5 A+ PT A L A2 A+ PT A2 5 0 00"68 123"3383 A+ PT A2 5 0 214"02
L,% B
RB2 5 R A2 > A %&&(' RB2 5 $8"887 = 16 RB2 5 42"887
D,' (* , '% %
T&% *% ',% ,&,(*'%& % '% &H,( , '% ',% A H K*, (-,&,-, ,' *- !
16
Sen( 60
ο
)
5
Sen( 30
5 "2376
5
(16) 2 + ( ) 2
5 18"47$2 D,' (* , '% %
ο
)
T
RB2 5
LB2 5
∆2 5 48"2230 2
tan
π R∆ ο
180
5 100"80
A+ PTB2 5 A+ PT B1 LB2 A+ PTB2 5 0 073"348 "2376 100"80
A!s PTB* - 10 2 )8/),0/ $ PROBLEMA 3.)6 Datos: A,+ , '% (/&%( %% , '% !()*&% 3.3" +, ,: C&,%%+ , A C&,%%+ , B
Ca"%u"ar:
5 N: $28" E: 416 5 N: 62$" E: $30
L% ,*%( , ,%',.
Figura 3.+3 Pro!"#$a 3.)6 So"u%i&' D(+-%(%
AB
5
( 530 − 416) 2 + ( 625 − 52) 2 5 14"623
T%),-, V% N. 1 5 T 5 178 -% AP 5
14"683 > 73"73 5 7$"$3
75,953
R2 5 L1 5
45 5 73"73 2
7,6 5 7"21$ 2
tan
π (178 )( 45 ) ο
180
5 13"80
L1 5
π ( 97,215 )( 76 ) ο
180
5 128"$1
A+(+% ,' PT *&% N. 1 ,%', 5 4 70 13"800 128"$1 A+(+% ,' PT *&% N. 1 ,%', 5 $ 238"7$2 149,683
(
Sen 59
)
5
Ac
A 5 16"438 73"73
Sen( 76)
A 5 243"168
243,168
R5
121 5 137"$78 2
tan
L1 *&% N. 2 5
π (137 ,578 )(121) 180
ο
5 20"$44
A+(+% PT1 " V% N. 2 5 $ 260"$44
PROBLEMA 3.) Datos: A((%',-, % '% (/&%( %% , '% !()*&% 3.4" +, ,: C&,%%+ , A 5 N: 426" E: 342 C&,%%+ , B 5 N: 200" E: $00 A+(+% , C 5 1 80 A+(+% , B 5 2 20 C*,&%+ 5 5 10
Ca"%u"ar: %@ L% ,*%( , ,%', ,-&, '%+ + %+. @ L% %+(+% ,' *- D.
Figura 3.+/ Pro!"#$a 3.) So"u%i&' ER 5 5 V. V 5 40 ?13600 +,)@ ?10001M@ 5 11"11 + 5 ?11"11 +@?1$ +,)@ 5 167 ER 5 167 <1 5 180 = θ 5 126;$29 <2 5 180 = α 5 40;369
126ο 52' T1 5 $0 -% 5 100"00 2
40ο 36' T2 5 $0 -% 5 18"$0 2
S, *,, (+,%& + *&%+ , ,' (+ +,-( &%( ( +, *,, %'(%& ,' &%( %&% %H& (%. R 5 120
126ο 52' T1 5 120 -% 5 240 2 40ο 36' T2 5 120 -% 5 44"40 2 A+ PC1 5 A+ POT S
S 5 4472"14 > T 1 5 4232"14
A+ PC1 5 0 000 4232"14 A+ PC1 5 4 232"14
A+ PT1 5 A+ PC1 L1
C ( ) 2 120
G 5 2 %&+,
G 5 4;469
ο
L1 5
10 126 52' ο
4 46'
A+ PT1 5 4 232"14 266"1$ A+ PT1 5 4 48"2 E-&,-%),(% PT 1 . PC2 5 2$10"68
A+ PC2 5 A+ PT 1 PT1 PC2 A+ PC2 5 4 48"2 2$10"68 A+ PC2 5 7 008"77 A+ PT2 5 A+ PC2 L2
L1 5 266"1$
C 2( 40)
G 5 2 %&+,
L2 5
(
)
ο
10 40 36' ο
4 46'
G 5 4;469
L2 5 8$"17
A+ PT2 5 7 008"7 8$"17 A+ PT2 5 7 04"14
<3 5 180; = β 5 20;$79
20ο 57' T3 5 120 -% 5 22"1 2 A+ PC3 5 A+ PT 2 PT2 PC3 A+ PC3 5 7 4"14 3030"72 A+ PC3 5 10 124"86
A+ PT3 5 A+ PC3 L3
C 2( 40)
G 5 2 %&+,
L3 5
(
ο
10 20 57'
)
ο
4 46'
A+ PT3 5 10 124"86 43"$ A+ PT3 5 10 168"81
A+ P! 5 A+ PT 3 302"4$ A+ P! 5 10 168"81 302"4$ A+ P! 5 13 18"26 <4 5 28;29
G 5 4;469
L3 5 43"$
28ο 2' T4 5 120 -% 5 2"6 2 C ( ) 2 40
G 5 2 %&+,
G 5 4;469
A+ PC4 5 A+ P A Q
Q 5 3640"0$ > T 4
A+ PC4 5 0 000 3610"0 A+ PC4 5 3 610"0
A+ PT4 5 A+ PC A L4
L4 5
(
ο
10 28 2'
)
ο
4 46'
L4 5 $8"81
A+ PT4 5 3 610"0 $8"81 A+ PT4 5 3 668"0
<$ 5 161;1$9
161ο 15' T4 5 120 -% 5 726"83 2 3250
R5
102ο 5' 74ο 3' 5 1633"17 + tan 2 2
tan
D+ *&%+ &%(+ ()*%',+ R 1 5 R2
102ο 5' T6 5 1633"17 -% 5 202$"13 2 ο 74 5' T7 5 1633"17 -% 5 1224"87 2
R( R&,%'
Sir(#
T6 T7 5 32$0
A+ PC6 5 A+ PT 4 PT4 PC6 A+ PC6 5 3 668"0 1$1"37 A+ PC6 5 $ 188"27
A+ PCC 5 A+ PC6 L6
10 3266,34
G 5 2 %&+,
L6 5
(
ο
10 402 5'
)
ο
0 21'
G 5 0;219
L6 5 216"67
A+ PCC 5 $ 188"27 216"67 A+ PCC 5 8 104"4
L7 5
(
ο
10 74 3'
)
ο
0 21'
A+ PT 5 A+ PCC L 7 A+ PT 5 8 104"7 211$"71 A+ PT 5 10 220"68
A+ PC$ 5 A+ PT PT.PC $ A+ PC$ 5 10 220"68 $32"41 A+ PC$ 5 1$ $4"04 A+ PT$ 5 A+ PC$ L$
L7 5 211$"71
L$ 5
(
ο
10 161 15'
)
ο
4 46'
L$ 5 338"20
A+ PT$ 5 1$ $4"0 338"20 A+ PT$ 5 1$ 887"38
A+ P! 5 A+ PT $ V
V 5 70$"6 > T $
A+ P! 5 1$ 887"38 7178"86 A+ P! 5 23 066"24
E%ua%i&' 7# E$a"$# - 1)3 2 )+8* - 1*3 2 0*/ E-&,-%),(%+ PT1 PC2 5 2$10"68 PT2 PC3 5 3030"72 PT4 PC6 5 1$1"37 PT PC$ 5 $328"41 C+ θ 5
HG. HI HG HI
HG. HI
C+ θ 5
HG HI
G 5 4000( > 2000 I 5 2$00( 12$0 C+ θ 5
( 4000" − 2000 ! )( 2500" + 1250 ! ) 4472,14 2795,03
θ 5 $3;79
<1 180; = θ 5 126;$29
I 5 2$00( > 12$0 I 5 3000( 7$0 C+ α 5
( − 2500" − 1250 ! )( 3000" − 750 ! ) 2795,08 3092,33
α 5 13;249
<2 5 180; = α 5 40;369
JI 5 =3000( 7$0 J! 5 2$00( = 17$0 C+ β 5
( − 3000" − 750 ! )( 2500" − 1750 ! ) 3092,33 3051,64
β 5 1$;39
<3 5 180; = β 5 20;$79
BA 5 =3$00( 1000 BC 5 3$00( = 7$0 C+ γ 5
( − 3500" − 1000 ! )( 3500" − 750 ! ) 3640,051 3579,46
γ 5 1$1;$89 <4 5 180; = γ 5 28;29
ED 5 7000( = 2000 E! 5 6$00( = 4$00
( 7000" − 2000 ! )( 6500" − 4500 ! )
C+
φ 5
φ 5
18;4$9
7280,11 7905,69
<$ 5 180; =
φ 5
161;1$9
CD 5 0( = 32$0 CB 5 =3$00( = 7$0 C+
( − 3500" − 750 ! )( 0" − 3250 ! )
Ω 5
3579,46 3250
Ω 5 77;$$9 <6 5 180; =
Ω 5 102;$9
DC 5 0( 32$0 DE 5 =7000( = 2000 C+ λ 5
( 0" − 3250 ! )( 7000" − 2000 ! ) 3250 7280
λ 5 10$;$79
<7 5 180; = λ 5 74;39
PROBLEMA 3.),
Datos: L+ K*, %%&,, , '% !()*&% 3.$.
Ca"%u"ar: L% ,*%( , ,%', , '% % 2 , '% % 1.
Figura 3.+6 Pro!"#$a 3.), So"u%i&' Para "a 9a No. ) <1 5 162; = 108; 5 $4; <2 5 162; = 41; 5 121; <3 5 ?360; = 312;@ 41; 5 8;
<1 ?V% N. 2@ 5 ?360; = 312;@ 108; 5 1$6;
Cur(a No. ) #' "a 9a No. ) 54 5 1"871 2
T1 5 3 -% L5
π ( 59 )( 54 ) ο
180
5 $$"606
P%&% '% % N. 2
PC1 5 0 00 PT1 5 0 00 $$"606 PT1 5 0 $$"606
Cur(a * #' "a 9a No. ) 121 5 61"862
T2 5 3$ -% CB
5 61"862 28 5 8"862
89,862
sen( 54
ο
AB
2
)
5
AB Sen( 67ο )
5
AC Sen( 59
ο
)
5 102"246
A+(+% ,' PI A 5 0 1"871 A+(+% ,' PIB 5 0 1"871 102"246 A+(+% ,' PIB 5 0 60"2$2
L *&% N. 2 5
π ( 35 )(121) ο
180
5 73"1$
A+(+% PT2 5 0 60"2$$ 73"1$ 5 1 034"170
L *&% N. 3 5
π ( 28 )( 89 ) ο
180
5 43"444
PT *&% N. 3 5 1 077"664
R%( %&% '% % N. 2 , /*( , '% -%),-,
( 89,862) ( Sen( 59
ο
AC 5
ο
Sen54
) ) 5 $"210
T%),-, 5 $"210 1"871 5 11$"081 T
R1 5
L5
115,081
∆ 2
tan
π ( 24,461)(156 ) 180
ο
5
156 5 24"461 2
tan
5 66"6
A!s%isa PT 9a No. * - 10 2 + PROBLEMA 3.)8 Datos: A((%',-, % '% (/&%( %% , '% !()*&% 3.6" +, ,: C&,%%+ , B 5 N: 4$.430" E: 32$4.210 C&,%%+ , B 5 140.240 A+(+% , C 5 P*- C A+(+% , B 5 2 20 C*,&%+ 5 5 $ ?&(,&% *&%@ H 10 ?+,)*% *&%@
Ca"%u"ar:
Figura 3.+ Pro!"#$a 3.)8
L%+ &,%%+ ,' *- P , %+(+% 4 640
So"u%i&' D(+-%(% BD 5 140"240 P*- ,( BD 5 P*- C C*,&% 5 5 $"0 ?&(,&% *&%@ H 10 ?+,)*% *&%@
T%),-, , '% &(,&% *&% 5 BC 5 70"12 70,12
R1 5
106 5 $2"83 2
tan
G1 5 2%&+,
L1 5
C 1 2 R1
( 5)(106) ο
5 25'25' '
5 2%&+,
5 2( 52,839)
5 $;2$92$99
5 7"721
PT1 5 4 3$7"420 7"74 5 4 4$$"141
A+(+% ,' PC2 5 4 4$$"141 70"12 A+(+% ,' PC2 5 4 $2$"261
G2 5 2%&+,
C 1 2 R1
5 2%&+,
10 2( 59)
5 ;4392299
L2 5 4 $2$"261 > 4 640 5 114"73 L2 5
G1 5
C 2∆ 2 G2 C 2 ∆ 2 L2
5
(
10 111 33'24' ' 114 ,739
)
5 $;2$92$99
C&,%%+ ,' *- D P&H,(,+ %(% %' *- D N 5 110"$11 N 5 86"340 C&,%%+ , D N 5 $10$"41 E 5 3167"870 A(*- %(% ,' *- P <2 5
(
) 5 111;339299
ο
114,739 9 43'22' ' 10 ο
C' ?2@ 5 2 ?$@
111 33'29' ' 2
5 7"$71
D,/',( & ,-& %&% *,&% , 10 10 5 3 6; 5 3 ;4392299 5 2;1090699 114"73 10 5 3346"74$ 60 10 5 $$;4694$99
A(*- ,' *- P N 5 2"11 E 5 2"73 C&,%%+ ,' *- P 5 4 460 N 5 $10$"41 2"11 5 $18"8$2 E 5 3167"870 2"73 5 317"663
PROBLEMA 3.)+
Datos: A,+ , '% (/&%( %% , '% !()*&% 3.7" +, ,: D(+-%(% AB 5 23$
Ca"%u"ar: L%+ ,*%(,+ , ,%', ,,+%&(%+.
Figura 3.+, Pro!"#$a 3.)+ So"u%i&' E, N. 1 <1 5 62; <2 5 118; T1 5 83"214 T2 5 83"213 R1 5 130"414 R2 5 $0 GC 5 4"1380; G 5 11"4783; L 5 14"82$ L 5 102"8023 A+ PE 5 0 147"$47 A+ PT 5 0 102"802 E*%( , E%', 5 0 14.82$ ?V% 1@ 5 1 102"802 ?V% 2@
C*&% E, 3 < 5 118; T 5 78"2211 R 5 47 G 5 12"2137; L 5 6"6128 A+ PE 5 0 06"6128 E*%( , E%', 5 0 06"6128 ?V% 3@ 5 0 176"3660 ?V% 2@ C*&% 4 V(% 2 <1 5 62; T 5 $2"274 R 5 87 G 5 6"$04; L 5 4"011 A+ PC 5 0 202"3131 A+ PE 5 0 26"404 E, 4 5 130"46 0 130"46
0 26"404 ?V% 2@ 5 0 130"46 ?V% 4@
PROBLEMA 3.*0 Datos: A,+ , '% (/&%( %% , '% !()*&% 3.8" +, ,: C&,%%+ , A C*,&%+
5 N: $000" E: 8000 5 5 10 ?& ,' E, 1@ H $ ?& ,' ,, 2@
Ca"%u"ar: %@ L%+ %+(+%+ P & ,' E, 1 H & ,' E, 2. @ L%+ C&,%%+ ,' P*- P.
Figura 3.+8 Pro!"#$a 3.*0 So"u%i&' <1 5 167; = 147; 5 6$; <2 5 180; = ?212; = 167;@ 5 13$; T %% *&% H '%+ (( , 2 %'' θ %'(% '% ',H , +, 2 5 %2 2 > 2% C+ θ
a 2 + b 2 − c 2 θ 5 C+ 2ab =1
802 + 802 − 402 θ 5 C+ 2 80 80 ( )( ) =1
C
G 5 2%&+,
2 R
5 2%&+,
10 2( 80)
5 7;9$"4299
<3 5 <1 = θ <3 5 36;2941"199 ο
L3 5
36 2'41,91' ' ο
7 9'59,42' '
L3 5 $0"2$
θ 5 28;$7918.099
A+ D ?E, N. 1@ 5 A+ PC 1 L 5 1 000 $"2$ A+ D ?E, N. 1@ 5 1 0$0"2$
802 + 402 − 802 5 7$;31920"699 <4 5 C+ 2 80 40 ( )( ) =1
5
C
G 5 2%&+,
L4 5
∆C GC
2 R
5 2%&+,
2( 40 )
75 31'20,96( 5)
5 7;9$"4299
ο
5
ο
7 9'59,92' '
L4 5 $2"64
A+ D ?E, 2@ 5 A+ PC 2 L A+ D ?E, 2@ 5 2 000 $2"64 A+ D ?E, 2@ 5 2 0$2"64
C&,%%+ , P %''%+ &,%%+ , PC 2
135 2
T1 5 40 -%
T1 5 6"$2
N PC2 5 $000
± 6"$7 C+ 212 5 418"11
E PC2 5 8000
± 6"$7 S, 212 5 748"83
NP 5 418"1 30 5 448"1 EP 5 748"83 38"73 5 787"$6 D 5 40 C+ 7$;31920"699 D 5 10
A 5 40 > 10 5 30 # 5 40 S, 7$;31920"699 # 5 38"73
PROBLEMA 3.*) Datos: P%&% '% /()*&% 3." %((%',-, +, -(,,: P'2.P'1 R%( %' P' 1 C*&%-*&% *&% R 2 T%),-, %' P' 3 C*,&%+
Ca"%u"ar:
5 88.460 5 R1 5 71.680 5 GC2 5 6; 5 T3 5 $$.00 5 1 5 2 5 3 5 10
Figura 3.++ Pro!"#$a 3.*)
L% ,*%( , ,%', ,' E, 3 , ,' E, 2.
So"u%i&' A+ PC2 5 600"$30 E, 1 A+ PC2 5 0 000
E, 2
A1 5 143;2$9 A2 5 12;$39 A3 5 24;1$9 A+ < 5 70 E, 1 A+ < 5 0 000 E, 3 <1 5 4;229 L1 5
C ∆1 Gc1
5 61"83
<3 5 180; = ?A 3 > A1@ 5 74;109 <2 5 A3 > A2 5 $6;229 P%&-(, , '% *&% N. 2 C
G2 5 2%&+,
2 R
5 6;
L2 5 10 D,+,% R2 5 $"$4 T2 5 R2 -%
∆2 2
T2 5 $1"1
A+ PT3 5 A+ PC2 L2 PI1 PI3 T3 ?E, N. 2@
L2 5
∆2 Gc2
5 $"44
PI 2 PI 3
5 PI2 PI3 > T2
A+ PI3 5 0 67"72 5 0 16"74 PI 2 PI 3
L3 5
6"88 > $1"16 5 18"6
∆3 Gc3
5 4"24
T 3
R3 5
∆3 5 72"84 2
tan
C
G3 5 2%&+,
2 R
5 7"87;
<1 PT1 5 A+ < = A+ PT 1 ?E, 1@ <1 PT1 5 A+ PC2 T2 PI1 PI2 > T1 L1 ?E, 1@ T1 5 72"68 -%
∆1 2
5 33"02
A+ PT1 5 13"2 <1 PT1 5 21"01
PROBLEMA 3.** Datos: P%&% '% !()*&% 3.100" %((%',-, +, -(,,: C&,%%+ , A D(+-%(% AB
5 N: $00" E: 300 5 38
Ca"%u"ar: L%+ %+(+%+ ,' *- , (-,&+,( P , '% V% 1 '% % 2.
Figura 3.)00 Pro!"#$a 3.** So"u%i&' P&(,& %'' '+ ,',,-+ , '% *&% N. 1 <1 5 A -%). S%'(% > A T%). E-&%% <1 5 114; R1 5 $2 T1 5 R -%
∆1 2
G1 5 2%&+,
L1 5
∆1C Gc1
T1 5 $2 -%
5
%'' A+ PC
5 2( 52)
5 $;30940.899
(114 )( 5 ) ο
114
5 30'40.8' '
5 103"42
2
5 80"07
A+ PC 5 A+ A > T1 5 4 328"7$0 > 80"07 A+ PC 5 4 248"66 D(( L% *&% , + H %'*' '+ ,',,-+ , %% *% %'' <2 H <3 T&% *% ',% K*, *% ,' PC H PT" '*,) %'' θ H α θ α 5 180 = < 1 5 180 > 114 5 66;
C '%+ (+-%(%+ + ()*%',+ θ H α + ()*%',+ θ 5 α 5 33;
D(+-%(% B.PT 5 T 1 =
AB
5 80"07 > 33
D(+-%(% B.PT 5 T 1 =
AB
5 42"07
<3 5 %&+,
42,07 52
5 38;$892799
<2 5 <1 = <3 5 7$;1932"199 A&% -,) '+ '*'+ , '+ ,',,-+
Cur(a No. 3
Cur(a No. *
<3
5 38;$892799
<2
5 7$;1932"199
R3
5 $2
R3
5 $2
T3
5 18"40
T3
5 3"2
G3 5 G1 5 $;30940"899
G2 5 G1 5 $;30940"899
L3 5 3$"36 A&% %'*' '% %+(+% & ,' ,, 1 H 2
L2 5 68"07
A+(+% P ?E, 1@ 5 A+ PC L 2 A+(+% P ?E, 1@ 5 4 248"66 68"07
A!s%isa P E4# )5 - 1/ 2 3),3
A+(+% P ?E, 2@ 5 A+ B ?C = R@ A+(+% P ?E, 2@ 5 0 424"270 ?66"8 = $2@
A!s%isa P E4# *5 - 10 2 /3+) PROBLEMA 3.*3 Datos: P%&% '% !()*&% 3.101" %((%',-, +, -(,,: C&,%%+ , PI C&,%%+ , A C&,%%+ , B
5 N: $00.730" E: 413"60 5 N: 4$4.120" E: 361.40 5 N: 447.080" E: 442.880
Figura 3.)0) Pro!"#$a 3.*3 Ca"%u"ar: L% %+(+% ,' *- P & ,' E, 1.
So"u%i&'
R 5 6$ < 5 173; = 7$; 5 8;
∆ 2
98 2
T 5 6$ -%
T 5 R -%
G 5 2%&+,
10 2( 65)
5 8;4924"4399
(10)( 98)
L 5
T 5 74"77
ο
8 49'24.43' '
L 5 111"06
413,96 − 361,94 500,730 − 442,080
-%=1 β 5
β 5 44;69$8"$499
α 5 0; = 44;69$8"$499
α 5 4$;$391.4699
( 442,080 − 454,120) 2 + ( 422,880 − 361,941) 2
AB
5
AB
5 81"24
API 5
( 413,960 − 361,900) 2 + ( 500,730 − 454,120) 2
API 5 6"84
Senα AB
5
ο
Senθ
Sen45 53'1.46' '
API
81,24
θ 5 38;69$4.99 φ 5
180 = ?38;69$4.99 4$;$391.4699@
φ 5
6;094.4499
a 2 + b 2 − c 2 θ 5 %&+ 2ab
5
Senθ
69,84
652 + 652 − 81,24 2 θ 5 %&+ ( )( ) 2 65 65 θ 5 77;21911"199
<2 5 <1 = θ <2 5 8 = 77;21911"199 <2 5 20;38948"99
G 5 2%&+,
L1 5
10 2( 65)
5 8;4924"4399
(10) ( 20ο 38'48.9' ')
L1 5 23"60012
ο
8 49'24.43' '
A+ PT 5 A+ PC L 1 A+ PT 5 3 82"$76 111"06 A+ PC 5 4 03"636
A+ P 5 A+ PT > L 1 A+ P 5 4 03"636 > 23"600
A!s P - 1/ 2 0+6// PROBLEMA 3.*/ Datos: P%&% '% !()*&% 3.102" %((%',-, +, -(,,: C&,%%+ , P D(+-%(% PQ P# H QN + %&%','%+
5 N: 10000" E: $000 5 273
Ca"%u"ar: %@ L% ,*%( , ,%', ,-&, '+ + ,,+. @ L%+ &,%%+ ,' *- , %+(+%+ $ 100
Figura 3.)0* Pro!"#$a 3.*/ So"u%i&' P%&% ,' ,, B *&% N. 1 R 5 88 <1 5 120; T1 5 88 -%
( 60 )
G1 5 2%&+,
ο
5 116
T1 5 1$2"420 5 3;1$921"1799
120ο L1 5 ο 3 15 ' 21 , 17 ' '
L1 5 184"28
A+(+% , PC2 ?E, A@ A+ P T1
4 00 1$2"420
A+ PC2 5 $ 0$2"420
A+(+% , PC2 ?E, B@ A+ PC1 L1
0 200 184"280
A+ PC2 5 0 384"280
E*%( , ,%', ,' ,, B , ,' ,, A
10 2 38/*80 E4# B5 - 16 2 06*/*0 E4# A5 P%&% ,' ,, A *&% N. 2 < 5 60; T2 5 R2 5
P# =
T1
T2 5 273 > 1$2"420
120,58
R2 5 208"8$1
ο
tan 30
G2 5 2%&+,
5 417,702
5 1;22918"299
LP 5 $ 100 > $ 0$2"420
T 2 5 120"$80
( 47,580) (1ο 22'18.2' ') 5
L P 5 47"$80
CL 5 2 ( 208,851) +, 6ο 31'55,97' '
C&,%%+ PC 2 N 5 1000 1$2"420 + 64;5 10066"817 E 5 $000 1$2"420 +, 64;5 $136"4
CL 5 47"478
C&,%%+ , E
N - )008), 2 /,/,8 s#' )+;*8<*/03<< - )008*/6 E - 6)3++/ 2 /,/,8 %os )+;*8<*/03<< - 6)8),66 PROBLEMA 3.*6 Datos: P%&% '% !()*&% 3.103" %((%',-, +, -(,,: C&,%%+ , A
Ca"%u"ar:
5 N: 1000" E: $00
Figura 3.)03 Pro!"#$a 3.*6
%@ L% ,*%( , ,%', ,-&, ,' ,, B H ,' ,, A @ L%+ %+(+%+ ,' *- Q @ L%+ &,%%+ ,' *- Q
So"u%i&' %@ P%&% ,' ,, B *&% N. 1 R 5 60
< 5 0; G1 5 2%&+,
5 120
5 4;46933"1199
90° 4°46'33,71' '
L1 5
L1 5 4"220
P%&% ,' ,, A , '% C*&% N. 2 R 5 70 < 5 0; G2 5 2%&+,
5 140
5 4;0$936"3399
90° 4°05'36,37' '
L1 5
L1 5 10"32
A+ PT1 ?E, B@ 5 A+ PC 1 L1 A+ PT1 ?E, B@ 5 4 $0 4"220 A+ PT1 ?E, B@ 5 $ 044"220
A+ PT1 ?E, A@ 5 A+ PC 2 L2 20 A+ PT1 ?E, A@ 5 2 $0 10"32 20 A+ PT1 ?E, A@ 5 3 07"32
E%ua%i&' 7# E$a"$# - 16 2 0//**0 E4# B5 - 13 2 0,++3* E4# A5 @
40 70
θ 1 5
%&+
θ 1 5
$$;090"3499
∆$ 2
5 17; 2$92"8399
CL 5 140 S, ?17;2$92"8399@ 5 41"24
55°09'0,34' ' 4°05'36,33' '
L5
L 5 67"364 5
#
A+ Q ?E, A@ 5 A+ PC 2 L A+ Q ?E, A@ 5 2 $0 67"364
A!s = E4# A5 - 13 2 0), 3/ %# 5 #
∆$ 2
S,
A+ Q ?E, B@ 5 A+ PC 1 60
%# 5
41"24 S, 17;2$92"8399
%#
A+ Q ?E, B@ 5 4 $0 60 12"$$4
A!s = E4# B5 - 16 2 0***6/ @ -% θ 2 5
12,554 30
5 (12,554) 2 + ( 30) 2 θ 5 210; = 180; 5 30; A#
θ 2 5 22;42927"6799 A#
5 32"$21
3
θ 4 5 30; = θ 2 θ 4 5 30; = 22;42927"6799
C&,%%+ , Q
N - )000 > 3*66* Cos ,;),<3*33<<5 - +,,/* E - 600 > 3*66* S#' ,;),<3*33<<5 - /6+*,*
PROBLEMA 3.* Datos: P%&% '% !()*&% 3.104" %((%',-, +, -(,,: C*&% , ,-& O 1 C*&% , ,-& O 2 C*&% , ,-& O 3 C*&% , ,-& O 4 C*&% , ,-& O $
5 R1 5 $2 5 R2 5 32 5 R3 5 20 5 R4 5 42 5 R$ 5 64
Figura 3.)0/ Pro!"#$a 3.* Ca"%u"ar: L%+ ,*%(,+ , ,%', ,,+%&(%+.
So"u%i&' E4# / L5
C ∆ G
5
(10 )( 90 ) 8°57 '41,75' '
5 100"43
G 5 2%&+,
10
5 8;$7941"7$99
2( 64)
2 000 100"43 5 0 100"43
E4# 3 T$ 5 R -%
∆
T$ 5 64 -%
2
90 2
T$ 5 64
1 000 2$ 64 5 1 08"00 E*%( , E%', 5 0 100"43 ?E, 4@ 5 1 08"00 ?E, 3@
E4# ) L5
C ∆ G
5
(10)( 90) 13°40'27,42' '
G 5 2%&+,
10
5 6$"81
5 13;40927"4299
2( 42)
2 000 63"81 5 0 06$"81
E4# * L5
C ∆ G
5
(10 )( 90 ) 11°2'27, 42' '
G 5 2%&+,
10 2( 52)
5 81"$$
5 11;297"699
0 000 T $ 1$ L1 5 0 000 64 1$ 81"$$ 5 0 160"$$ L2 5
C ∆ G
5
G2 5 2%&+,
(10)( 90 ) 17°58'42,95' ' 10 2( 82)
5 $0"0$
5 17;$8942"$99
0 160"$$ $0"0$ 5 0 210"61
L3 5
C ∆ G
5
G3 5 2%&+,
(10 )( 90) 28°57'18,09' ' 10 2( 20)
5 31"08
5 28;$7918"099
0 210"61 31"08 5 2 241"6
E%ua%i&' 7# E$a"$# - 1* 2 */)+ E4# *5 - 10 2 068) E4# )5 PROBLEMA 3.*, Datos: P%&% *% *&% (&*'%& +(', +, -(,,: A+(+% ,' PC R%( , '% *&% D,/',( &((%' C*,&% *(%
5 O426.700 5 R560.170 5 < 5 $0 ;D 5 5 10
Ca"%u"ar: L% *&% & ,' - , '%+ &%',+ +&, '% -%),-," , -%' %,&% K*, +, -,)% '+ (++ *-+ , '% *&% ,/',-%+ ,+, ,' PC & ,' - , '%+ ,/',(,+ H *,&%+.
So"u%i&' R 5 60"170 < 5 $0 5 10
50 5 28"0$8 2
T 5 60"170 -%
G3 5 2%&+,
10 2( 60,170)
5 ;319$"299
L 5
10( 50 ) 9°31'59,92' '
5 $2"448
A+ PT 5 A+ PC L A+ PT 5 46"700 $2"448 A+ PT 5 0 47"148
P& ,' ,- , '%+ ,/',(,+ C 5 10 " '%+ %+(+%+ +, %'*'% %% 10 P%&% %''%& '% &(,&% ,/',( -,)" 430 > 426"70 5 3"3 G2 5 4;4$9$"699 5 1;34922"80 P&(,&% ,/',( S*% G 2 % '% ,/',( %-,&(& +, -(,, '%+ ,/',(,+ , '%+ %+(+%+ %% 10 P%&% '% ,/',( ,' PT 47"148 > 470 5 "148 5 4;21937"399 L +* % '% ,/',( , 0 470 %&% -,,& '% ,' PT P& ,' - , '%+ &%',+ +&, '% -%),-, D,' '(& D(+, G,-&( , V%+ , J%,+ C%&,%+ G&(+%',+ ?P%) 3$" O-&+ -+ , %'*' H '%'(%( , *&%+ (&*'%&,+ +(',+@ +, -% '%+ +()*(,-,+ /&*'%+ 5
R(1 − Cos 2δ ) tanδ
F 5 R (1 − Cos2δ )
N-% '%+ &,+*,+-%+ +, *,+-&% , '% +()*(,-, -%'%
Cart#ra 7# I7#a"i?a%i&' 7# u'a %ur(a %ir%u"ar or #" $@to7o 7# "as 'or$a"#s so!r# "a ta'g#'t# ESTACIN
ABSCISAS
PC
O426.700 430 440 4$0 460 470 O47.148
PT
DEFLEIONES δ 00=00=00.00 01=34=22.80 06=20=22.76 11=06=22.72 1$=$2=22.68 20=38=22.64 2$=00=00.0$
$5
$5
0.000 3.2 13.207 22.747 31.6$ 3.66 46.03
0.000 0.01 1.467 4.46$ .002 14.$2 21.43
PROBLEMA 3.*8 Datos: P%&% '% +(-*%( %% , '% !()*&% 3.10$" +, -(,,: <5100;D
β 521;
PI.P52$
Ca"%u"ar: E' &%( , '% *&% K*, %+% & ,' *- P.
Figura 3.)06 Pro!"#$a 3.*8
So"u%i&' P& '% ,*%( 3"18 ,' '(& J%,+ C&,%+ 1 − Cosθ α 5 %&-% ∆ − tan Senθ 2 1 − Cosθ -% 21 5 %&-% ∆ − Senθ tan 2
D,+,%+ θ
θ 5 38"$64
C '% ,*%( 3"1 ,' '(& J%,+ C&,%+ ,+,%+ ,' &%( 2
∆ 2$ 5 R tan − senθ + (1 − Cosθ ) 2
R - /)0+ $ PROBLEMA 3.*+ Datos: P%&% *% *&% (&*'%& +(', +, -(,,: A+(+% ,' PC D,/',( &((%' G&% , *&%-*&% C*,&% *(%
5 4$23.800 5 < 5 70 ;D 5 G C 5 6;30 5 5 $
Ca"%u"ar: L%+ ,/',(,+ ,+, ,' PC H ,+, ,' PI.
So"u%i&' Cl
R1 5
G 2Sen 2
R1 5
5 2( 0,056)
R1 5 44"0
T 5 R -%
L 5
C ∆ G
∆ 2
T 5 30"87
5 $3"84
A+ P1 5 4 $23"8 30"87 5 4 $$4"67 A+ PT 5 A+ A+ PC L L 5 4 $76"646 $76"646
C%'*' , D,/',(,+ P& *% &,)'% -,,+ 5 0;4694899 D,/',( % 1"2 %&% '',)%& % '% %+(+% , 4 $2$ D,+, 4 $2$ %+-% %' 4 $7$ +, +*% G2 5 3;1$9 *% % *% F %+ -,,+ '% %&-,&% , ,/',(,+ K*, +, +-&%&% , '% +()*(,-, -%'%"
CARTERA DE DEFLEIONES E+-%( PC
PROBLEMA 3.30 Datos:
A+(+%+ 4 $23"8 4 $2$ 4 $30 4 $3$ 4 $40 4 $4$ 4 $$0 4 $60 4 $6$ 4 $70 4 $7$ 4 $77"646
D,/',( 00;0090099 00;4694899 04;0194899 07;1694899 10;3194899 13;4694899 17;0194899 20;1694899 23;3194899 30;0194899 33;1694899 3$;0090099
D, *% *% *& *&% % (&* (&*'%& '%& * *,+,+-% % , + + &%(+ &%(+ +, , , '+ +()*(, +()*(,-, -,+ + ,',,-+: A+(+% ,' PI 5 1002.160 1002.160 D,/',( &((%' 5 < 5 68;32$4 D R%( , '% &(,&% *&% 5 106.680 R%( , '% +,)*% *&% 5 1$2.400 D,/',( , '% &(,&% *&% 5 40 ;1834
Ca"%u"ar: %@ L%+ -%),-, -%),-,+ + '%&)% '%&)% H &-% , '% *&% *,+-%. *,+-%. @ L%+ %+(+%+ %+(+%+ ,' ,' PC" PCC PCC H PT *+% '% ,/((( ,/((( & %&.
Cart#ra 7# I7#a"i?a%i&' 7# u'a %ur(a %ir%u"ar 7#s7# #" PC 7#s7# #" P" DEFLEIONE DEFLEIONE ESTACI S DESDE EL NGULO PI.P ABSCISAS S DOBLES α N PC $5 ϕ PC
4$23.800 $2$ $30 $3$ $40 $4$ $$0 $$$ $60 $6$ $70 $7$ 4$77.646
PT
δ 00=00=00 00=46=48 04=01=48 07=16=48 10=3148 13=46=48 17=01=48 20=16=48 23=31=48 2646=48 30=0148 33=1648 3$=00=00
00=00=00 01=33=36 08=03=36 14=33=36 21=03=36 27=33=36 34=03=36 40=33=36 47=03=36 $3=33=36 60=03=36 66=33=36 70=00=00
00=00=00.00 00=01=$3.60 01=00=38.08 04=0$=34.26 11=0$=13.8 2$=32=0$.22 $0=4$=10.24 78=1$=11.21 $=42=27.3$ 104=24=11.30 108=22=12.0 10=$0=1.40 110=00=00.00
So"u%i&' %@ TC 5
TC 5
R1
− R2Cos∆ − ( R1 − R2 ) Cos∆1 Sen∆
106,680 − 152,4Cos68°32'54' '−(106,68 − 152,4) Cos40°18'34' '
Sen68°32'54' '
30.877 2.677 24.68 1.842 1$.317 11608 .768 10.822 14.127 18.48$ 23.27$ 28.231 30.877
TC 5 2"16
TL 5
R2
TL 5
− R1Cos∆ − ( R1 − R2 ) Cos∆ 2 Sen∆
152 152,4 − 106 106,68Cos68°32'54' '−(106,68 − 152,4) Cos28°14'20' '
Sen68°32'54' '
TL 5 78"$48
@
PC 5 PI TL PC 5 1 002"16 > 78"$48 PC 5 0 23"612
L1 5
L2 5
π R1∆1 180
π R2∆ 2 180
5 7$"0$28
5 112
PCC 5 PC L 1 5 0 23"612 7$"0$28 PCC 5 0 8"66$
PT 5 PCC L 2 PT 5 0 8"66$ 7$"112 PT 5 1 073"777
PROBLEMA 3.3) Datos: L% (+% (/&%( %% , ,' E,' 3.23.
Ca"%u"ar: L%+ -%),-,+ , ,-&%% H +%'(% , '% *&% *,+-% , -&,+ &%(+" *-('(% ,' - ),,&%' % & 1%+ ,&,+(,+ , '%+ ,*%(,+ ?3=2$@ H ?3=26@.
So"u%i&' ( T 3 + T 1 ) Sen∆1 Sen( ∆3 + ∆1 ) T T + + TE 5 T2 2 3 Sen( ∆3 − ∆1 ) Sen∆
( T 3 + T 1 ) Sen∆1 Sen( ∆ 2 ) ( T 3 + T 1 ) Sen∆ 2 T T + + TS 5 T1 2 3 + Sen( ∆ + ∆ ) ( ) Sen Sen ∆ + ∆ ∆ 3 1 3 1 α 5 ?180; = < 2 = β @ β 5 180 = <
α 5 180 = <2 > 180 <
ρ 5 180 = α 5 12 T1 5 R1 -%
T2 5 R2 -%
T3 5 R3 -%
∆1 2
∆
2
2
∆3 2
5 30
5 22"$
5 12"74
<3 5 180 180 > 30 = 2 <3 5 21
T3 5 12"74 <3 5 180 = < 1 = ρ <3 5 180 = < 1 = ?180 = α @ <3 5 180 = < 1 = ?< = <2@ 5 180 < = < 1 <2
PROBLEMA 3.3* Datos: P%&% *% *&% (&*'%& , -&,+ &%(+ +, ,: A+(+% ,' P' D,/',( &((%' D,/',(,+ (((*%',+ R%( , '% +,)*% *&% R%( , '% &(,&% *&% R%( , '% -,&,&% *&% C*,&%+
5 2422.020 5 < 5 84; 5 < 1 5 <2 5<3 5 R 2 5 $0 5 R 1 5 1.$R2 5 R 3 5 R1 5 1 5 3 5 10 " 2 5 $
Ca"%u"ar: %@ L%+ -%),-,+ , ,-&%% H +%'(%. @ L% %+(+% ,' PT , '% *&% *,+-%.
So"u%i&' R1 5 1"$ $0 5 7$ R2 5 $0 R3 5 R1 5 7$
TL 5 T%),-, L%&)% TL 5
R1 − R1Cos∆ + ( R1 − R3 ) Cos( ∆1 + ∆3 ) + ( R3 − R2 ) Cos∆ 2 Sen∆
TL 5
75 − 75Cos84° + ( 75 − 50 ) Cos( 28° + 28°)
+ ( 50 − 75) Cos 28°
°
Sen84
TL 5 $"32 5 T%),-, , S%'(%
CL 2 R 1
CL 5 10
10 2( 75)
G1 5 7;3894299
G1 5 2 %&+,
G1 5 2 %&+,
L5
CL∆1 G1
5
(10)( 28) 7°38'42' '
G2 5 2 %&+,
L 5
L 5
5 2( 50 )
( 5)( 28) 5°43'55' '
G2 5 2 %&+,
5 36"62$
5 24"422
10 2( 75 )
(10)( 28) 7°38'42' '
G2 5 $;439$$99
G2 5 7;3894299
5 36"62$
A+(+% ,' PC 5 2 422"02 > $"32 5 2 236"628 A+(+% ,' PT 5 2 236"628 36"62$ 24"421 36"62$
A!s%isa 7#" PT - 1* 2 /030* PROBLEMA 3.33 Datos: P%&% '% !()*&% 3.106" +, -(,,:
C*&% , ,-& O 1 5 R1 5 60 C*&% , ,-& O 2 5 R2 5 40 C*&% , ,-& O 3 5 R3 5 30
Ca"%u"ar: %@ L% %+(+% , B +&, ,' *,-, H '% , B ,% ,' *,-,. @ L% ,(,-, *(/&, , '% ',% K*, % ,+, ,' *- B ?+&, ,' *,-,@ %+-% ,' *- B ?,% ,' *,-,@" +( ,&-(%',-, ,+-+ + *-+ ,+- +,%&%+ 7 ,-&+.
Figura 3.)0 Pro!"#$a 3.33 So"u%i&' %@ R1 5 60 R2 5 40 R3 5 30
P%&% '% *&% N. 1
< 1 5 ?360; = 324;@ $3; 5 8;
P%&% '% *&% N. 2
< 2 5 ?141; = $3;@ 5 88;
P%&% '% *&% N. 3
< 3 5 ?232; = 141;@ 5 1;
E-,+
∆1 2
T2 5 R2 -%
T1 5 60 -%
89 2
T1 5 $8"62
T1 5 R1 -%
L1 5
L1 5
π R1∆1 180
π ( 60 )( 89 ) 180
L1 5 3"201
T3 5 R1 -%
T2 5 40 -%
88 2
T3 5 60 -%
T2 5 38"627
T3 5 30"$28
L2 5
L2 5
π R2∆ 2 180
π ( 40)( 88) 180
L2 5 61"436
91 2
L3 5
L3 5
π R3∆ 3 180
π ( 30)( 91)
A+(+% ,' P*- B9 5 2 800 3"201 61"436 47"647 70 A+(+% B9 5 3 072"284 C&,%%+ ,' PI 1 ?P%&% '% C*&% N. 1@ A(*- 5 324; D(+-%(% 5$8"62 A+*(, *%+ &,%%+ ,
E 5 100"00
180
L3 5 47"647
C ,+- -,,+ K*,"
N 5 100"00
∆3 2
∆2 2
E-,+ &,%%+ ,' PI 1 N 5 147"701 E 5 6$"343
C&,%%+ ,' PI 2 ?P%&% '% C*&% N. 2@ A(*- 5 $3; D(+-%(% 5 7"$8 N 5 206"432 E 5 143"281
C&,%%+ ,' PI 3 ?P%&% '% C*&% N. 3@ A(*- 5 232; D(+-%(% 5 6"1$$ N 5 1$2"688 E 5 186"802
C&,%%+ ,' B9 A(*- 5 232; D(+-%(% 5 100"$28 N 5 0"76 E 5 107"$8$ D(+-%(% , B % B9 BB ' 5 BB ' 5
(100 − 90,796) 2 − (100 − 107,585) 2
11"26
