EJEMPLO EJERCICIO METODOS NUMERICOS

Un ingeniero civil involucrado en la construcción, requiere 4800, 5810 y 5690 metros cúbicos de arena, de ripio y piedras, respectivamente, para un proyecto de construcción. La composición de cada cantera es:

Arena Ripio Piedra

CANTERA1 52% 30% 18%

CANTERA2 20% 50% 30%

CANTERA3 25% 20% 55%

¿Cuántos metros cúbicos se debe transportar desde cada cantera para cumplir con las necesidades del Ingeniero?

MATRIZ

■(52&20&25@30&50&20@18&30&55) ■(4800@5810@5690)

■(52x1&20x2&25x3=&4800@30x1&50x2&20x3=&5810@18x1&30x2&55x3=&5690)

■(x1=&-20/52x2&-25/52x3&+4800/52@x2=&-30/50x1&-20/50x3&+5810/50@x3=&-18/55x1&-30/55x2&+5690/55)

X0 [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)

X1 [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)

X2 [59.6514, 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(82.6121)&+5690/55&=50.7235)

X3 [41.7692, 71.6121, 50.7235]

■(x1=&-20/52(41.7692)&-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)

X4 [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)

X5 [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)

X6 [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)

X7 [40.1155, 71.6303, 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)

X8 [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)

X9 [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)

X10 [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)

X11[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)

Ahora hacemos la comparación.

■(52(40.1163)&+20(71.6279)&+25(51.2558)&=4800@30(40.1163)&+50(71.6279)&+20(51.2558)&=5810@18(40.1163)&+30(71.6279)&+55(51.2558)&=5690)

K X1 X2 X3

0 0.0000 0.0000 0.0000

1 92.3077 60.8154 40.0727

2 49.6514 70.3801 48.8159

3 41.7692 71.6121 50.7235

4 40.3783 71.6836 51.1397

5 40.1507 71.6537 51.2305

6 40.1185 71.6367 51.2503

7 40.1155 71.6306 51.2546

8 40.1158 71.6287 51.2555

9 40.1161 71.6281 51.2558

10 40.1162 71.6280 51.2558

11 40.1163 71.6279 51.2558

12 40.1163 71.6279 51.2558

El Ingeniero Un ingeniero civil involucrado en la construcción, requiere 4800, 5810 y 5690 metros cúbicos de arena, de ripio y piedras, respectivamente, para un proyecto de construcción. La composición de cada cantera es:

Arena

...