Linear Programming Solved Problems

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Choose the scales so that the feasible region is shown fully within the grid.

(if necessary, draft it out on a graph paper first.) Shade out all the unwanted regions and label the required region It also possible to test the vertices of the feasible region to find the minimum or maximum values, instead of using the linear objective function.

b) Plot the inequalities on the Cartesian grid and show the region that satisfies all the inequalities.

Label the region ≤ 72 b) Write out the equations for the inequalities and draw them on the graph paper.

A farmer plans to mix two types of food to make a mix of low cost feed for the animals in his farm.

A bag of food A costs and contains 40 units of proteins, 20 units of minerals and 10 units of vitamins.

Let x be the number of bags of food A and y the number of bags of food B.

Cost C(x,y) = 10 x 12 y \[ \begin \ x \ge 0 \ \ y \ge 0 \ \ 40x 30y \ge 150 \ \ 20x 20y \ge 90 \ \ 10x 30y \ge 60 \ \end \] .

Vertices of the solution set: A at (0 , 0) B at (0 , 1429) C at (1333 , 667) D at (2000 , 0) Calculate the total profit P at each vertex P(A) = 2 (0) 3 ()) = 0 P(B) = 2 (0) 3 (1429) = 4287 P(C) = 2 (1333) 3 (667) = 4667 P(D) = 2(2000) 3(0) = 4000 The maximum profit is at vertex C with x = 1333 and y = 667.

Hence the store owner has to have 1333 toys of type A and 667 toys of type B in order to maximize his profit. It takes 2 hours to produce the parts of one unit of T1, 1 hour to assemble and 2 hours to polish.


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