Question 54944: THE ECONOMY OF A DEVELOPING COUNTRY IS BASED ON AGRICULTURAL PRODUCTS , INDUSTRIAL PRODUCTS AND MINING PRODUCTS . AN OUTPUT OF ONE TON OF AGRICULTURAL PRODUCTS REQUIRES AN INPUT OF 0.1 TON OF AGRICULTURAL PRODUCTS , 0.01 TONS OF INDUSTRIA PRODUCTS , 0.02 TONS OF MINING PRODUCTS . AN OUTPUT OF ONE TON OF MINING PRODUCTS REQUIRES AN INPUT OF 0.01 TON OF AGRICULTURAL PRODUCTS ,0.05 OF INDUSTRIAL PRODUCTS AND 0.13 OF MINING PRODUCTS . ONE TON OF INDUSTRIAL PRODUCTS REQUIRED AN INPUT OF 0.04 OF AGRICULTURAL PRODUCTS , 0.15 OF INDUSTRIAL PRODUCTS , AND 0.03 OF MINING PRODUCTS .
QUESTION :
1)Find the necessary gross productions to provide surplusses of 2300 tons agricultural , 3000 tons of industrial products and 4500 tons of mining products
2)distribute the gross production among derived and independent demands
Answer by jenrobrody(19)   (Show Source):
You can put this solution on YOUR website! I am not exactly sure of the context,definitions, or prerequistes of this problem but I shall try anyway.
A=number of tons of agricultural products
I=number of tons of industrial products
M=number of tons of mining products
To make 2300 tons of agricultural products we use the information given in the problem:
0.1A + 0.01I + 0.02M = 2300
The other two surpluses are set up in a similiar fashion:
0.01A + 0.05I + 0.13M = 4500
0.04A + 0.15I + 0.03M = 3000
There are three equations with three unknowns.
And there are many ways to solve. The standard algebraic method is to cancel out one variable with an addition method twice, then solve the remaining two variable/two equation.
The easiest way to solve is to use "computational power" and set up as maticies:
Let B equal a 3x3 matrix:
B = [0.1 , 0.01, 0.02;
0.01, 0.05, 0.13;
0.04, 0.15, 0.03]
and C equal a 3x1 matrix:
C = [2300;
4500;
3000]
To find your solution multiply the inverse of B with C:
(B^-1)(C)= (16100, 9784, 29610)
...which means that: To make 2300 tons of agric., 4500 tons of mining, and 3000 tons of industy, the country would need 16,100 tons agric., 9,874 tons industry, and 29,610 tons of mining.
I'm not entirely sure of what part 2 is asking, but again I try:
The 2300 tons of agric. product requires : 0.1A + 0.01I + 0.02M
Using the answers for A,I, and M from the above paragraph:
The 2300 tons of agric. product requires : 1,610 tons agric., 98.74 tons industry, 592.2 tons mining.
Likewise, 4500 tons of mining requires: 161 tons agric., 493.7 tons industry, and 384.9 tons mining.
3000 tons of industry requires: 624 tons agric, 1481.1 industry, and 888.3 tons mining.
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