1) Un ingeniero civil para realizar una construccion requiere 4800, 5810 y 5690 m^2 de arena, grava na y grava gruesa respectivamente, para un proyecto. Existen 3 minas de las cuales se pueden extraer los materiales. La composicion de estas canteras es: Cantera 1 Cantera 2 Cantera 3 Arena % 52 20 25 Grava na %

30 50 20 Grava gruesa % 18 30 55

Cuantos metros cubicos de cada material se deben extraer de cada mina para satidfacer las necesidades del ingeniero? Matriz 52 30 18 20 50 30 25 20 55 4800 5810 5690

52X1+ 0X2+ 5X3= 800 2 2 4 30X1+ 0X2+ 0X3= 810 5 2 5 18X1+ 0X2+ 5X3= 690 3 5 5 X1= -20/52X2-25/52X3+4800/52 X2= -30/50X1-20/50X3+5810/50 X3= -18/55X1-30/55X2+5690/55 X^0 [0,0,0] X1= -20/52(0)-25/52(0)+4800/52=92.3077 X2= -30/50(92.3077)-20/50(0)+5810/50=60.8154 X3= -18/55(92.3077)-30/55(60.8154)+5690/55=40.0727 X^1[92.3077,60.8154,40.0727] X1= -20/52(60.8154)-25/52(40.0727)+4800/52=49.6514 X2= -30/50(49.6514)-20/50(40.0727)+5810/50=70.3801 X3= -18/55(49.6514)-30/55(70.3801)+5690/55=48.8159 X^2 [49.6514,70.3801,48.8159] X1= -20/52(70.3801)-25/52(48.8159)+4800/52=41.7692 X2= -30/50(41.7692)-20/50(48.8159)+5810/50=71.6121 X3= -18/55(41.7692)-30/55(71.6121)+5690/55=50.7235 X^3 [41.7692,71.6121,50.7235] X1= -20/52(71.6121)-25/52(50.7235)+4800/52=40.3783 X2= -30/50(40.3783)-20/50(50.7235)+5810/50=71.6836 X3= -18/55(40.3783)-30/55(71.6836)+5690/55=51.1397

X^4 [40.3783,71.6836,51.1397] X1= -20/52(71.6836)-25/52(51.1397)+4800/52=40.1507 X2= -30/50(40.1507)-20/50(51.1397)+5810/50=71.6537 X3= -18/55(40.1507)-30/55(71.6537)+5690/55=51.2305 X^5 [40.1507,71.6537,51.2305] X1= -20/52(71.6537)-25/52(51.2305)+4800/52=40.1185 X2= -30/50(40.1185)-20/50(51.2305)+5810/50=71.6367 X3= -18/55(40.1185)-30/55(71.6367)+5690/55=51.2503 X^6 [40.1185,71.6367,51.2503] X1= -20/52(71.6367)-25/52(51.2503)+4800/52=40.1155 X2= -30/50(40.1155)-20/50(51.2503)+5810/50=71.6306 X3= -18/55(40.1155)-30/55(71.6306)+5690/55=51.2546 X^7 [40.1155,71.6306,51.2546] X1= -20/52(71.6306)-25/52(51.2546)+4800/52=40.1158 X2= -30/50(40.1158)-20/50(51.2546)+5810/50=71.6287 X3= -18/55(40.1158)-30/55(71.6287)+5690/55=51.2555 X^8 [40.1158,71.6287,51.2555] X1= -20/52(71.6287)-25/52(51.2555)+4800/52=40.1161 X2= -30/50(40.1161)-20/50(51.2555)+5810/50=71.6281 X3= -18/55(40.1161)-30/55(71.6281)+5690/55=51.2558 X^9 [40.1161,71.6281,51.2558] X1= -20/52(71.6281)-25/52(51.2558)+4800/52=40.1162 X2= -30/50(40.1162)-20/50(51.2558)+5810/50=71.6280 X3= -18/55(40.1162)-30/55(71.6280)+5690/55=51.2558 X^10 [40.1162,71.6280,51.2558] X1= -20/52(71.6280)-25/52(51.2558)+4800/52=40.1163 X2= -30/50(40.1163)-20/50(51.2558)+5810/50=71.6279 X3= -18/55(40.1163)-30/55(71.6279)+5690/55=51.2558 X^11[40.1163,71.6279,51.2558] X1= -20/52(71.6279)-25/52(51.2558)+4800/52=40.1163 X2= -30/50(40.1163)-20/50(51.2558)+5810/50=71.6279 X3= -18/55(40.1163)-30/55(71.6279)+5690/55=51.2558 Comparacion. 52(40.1163)+20(71.6279)+25(51.2558)=4800 30(40.1163)+50(71.6279)+20(51.2558)=5810

K 0 1 2 3 4 5 6 7 8 9 10 11 12

X1 0.0000 92.3077 49.6514 41.7692 40.3783 40.1507 40.1185 40.1155 40.1158 40.1161 40.1162 40.1163 40.1163

X2 0.0000 60.8154 70.3801 71.6121 71.6836 71.6537 71.6367 71.6306 71.6287 71.6281 71.6280 71.6279 71.6279

X3 0.0000 40.0727 48.8159 50.7235 51.1397 51.2305 51.2503 51.2546 51.2555 51.2558 51.2558 51.2558 51.2558

Para satisfacer las necesidades del ingeniero se deben extraer: 40.1163m^3 de arena 71.6279m^3 de grava na 51.2558m^3 de grava gruesa Cantera 1=cantera 2=cantera 3=13.3721m^3 Cantera 1=cantera 2=cantera 3=23.8760m^3 Cantera 1=cantera 2=cantera 3=17.0853m^3 2) El siguiente sistema de ecuaciones se disena para determinar concentraciones (las concentraciones estan en g/m^3) en una serie de reactores, como funcion de la cantidad de masa de entrada a cada uno de los reactores. 11C1 - 2C2 - 3C3=500 -5C1 - 21C2 - 2C3=200 -5C1 - 5C2 +22C3=300 Matriz: 11C1 - 2C2 - 3C3=500 -5C1 - 21C2 - 2C3=200 -5C1 - 5C2 +22C3=300 C1=2/17C2+3/17C3+500/17 C2=5/21C1+2/21C3+200/21 C3=5/22C1+5/22C2+300/22 X^0 [0,0,0] C1=2/17(0)+3/17(0)+500/17=29.4118 C2=5/21(29.4118)+2/21(0)+200/21=16.5266 C3=5/22(29.4118)+(16.5266)+300/22=24.0769

X^1[29.4118,16.5266,24.0769] C1=2/17(16.5266)+3/17(24.0769)+500/17=595.5650 C2=5/21(595.5650)+2/21(24.0769)+200/21=153.6180 C3=5/22(595.5650)+(153.6180)+300/22=183.9052 X^2 [595.5650,153.6180,183.9052] C1=2/17(153.6180)+3/17(183.9052)+500/17=79.9383 C2=5/21(79.9383)+2/21(183.9052)+200/21=46.0715 C3=5/22(79.9383)+(46.0715)+300/22=42.2750 X^3 [79.9383,46.0715,42.2750] C1=2/17(46.0715)+3/17(42.2750)+500/17=34.8319 C2=5/21(34.8319)+2/21(42.2750)+200/21=21.8433 C3=5/22(34.8319)+(21.8433)+300/22=26.5171 X^4 [34.8319,21.8433,26.5171] C1=2/17(21.8433)+3/17(26.5171)+500/17=36.6611 C2=5/21(36.6611)+2/21(26.5171)+200/21=20.7781 C3=5/22(36.6611)+(20.7781)+300/22=26.6907 X^5 [36.6611,20.7781,26.6907] C1=2/17(20.7781)+3/17(26.6907)+500/17=36.5664 C2=5/21(36.5664)+2/21(26.6907)+200/21=20.7721 C3=5/22(36.5664)+(20.7721)+300/22=26.6678 X^6 [36.5664,20.7721,26.6678] C1=2/17(20.7721)+3/17(26.6678)+500/17=36.5616 C2=5/21(36.5616)+2/21(26.6678)+200/21=20.7687 C3=5/22(36.5616)+(20.7687)+300/22=26.6660 X^7 [36.5616,20.7687,26.6660] C1=2/17(20.7687)+3/17(26.6660)+500/17=36.5609 C2=5/21(36.5609)+2/21(26.6660)+200/21=20.7684 C3=5/22(36.5609)+(20.7684)+300/22=26.6660 X^8 [36.5609,20.7684,26.6660] C1=2/17(20.7684)+3/17(26.6660)+500/17=36.5609 C2=5/21(36.5609)+2/21(26.6660)+200/21=20.7684 C3=5/22(36.5609)+(20.7684)+300/22=26.6660 X^9 [36.5609,20.7684,26.6660]

K 0 1 2 3 4 5 6 7 8 9

C1 0.0000 29.4118 595.5650 79.9383 34.8319 36.6611 36.5664 36.5616 36.5609 36.5609

C2 0.0000 16.5266 153.6180 46.0715 21.8433 20.7781 20.7721 20.7687 20.7684 20.7684

C3 0.0000 24.0769 183.9052 42.2750 26.5171 26.6907 26.6678 26.6660 26.6660 26.6660

Las concentraciones en funcin de la cantidad de masa entrante a cada reactor son: C1=36.5609 C2=20.7684 C3=26.6660