We are given inequalities : 5x+y=>10
x+y<=6
x+4y=>12
x=>0
y=>0 .
Let us graph it first and the find the coordinate of the vertices of feasible region.
Coordinates of the feasible region are (1,5) , (1.474, 2.632) and (4,2).
Now, we need to plug those cordinates (1,5) , (1.474, 2.632) and (4,2) in the given function c=10,000x + 20,000y.
Let us plug those points one by one.
For (1,5)
c=10,000x + 20,000y = 10,000(1)+ 20,000(5)= 10,000+ 100,000 = 110,000.
For (1.474, 2.632)
C = 10,000(1.474)+ 20,000(2.632) = 67,380.
For (4,2)
C = 10,000(4)+ 20,000(2) = 80,000.
We got the minimum value 67,380 for (1.474, 2.632) coordinate.
Therefore, minimum for given function C=10,000x + 20,000y is 67,380.