I set the following up as a linear programming problem and see how to get their dual, but I do not know what the dual of each gives me exactly. Like in problem 1, I get the max profit but what is the dual going to five me? Nor do I understand the significance of the dual in these two problems. Can someone please explain.

A company produces two types of tables, T1 and T2. It takes 2 hours to produce the parts of one unit of T1, 1 hour to assemble and 2 hours to polish. It takes 4 hours to produce the parts of one unit of T2, 2.5 hour to assemble and 1.5 hours to polish. Per month, 7000 hours are available for producing the parts, 4000 hours for assembling the parts and 5500 hours for polishing the tables. The profit per unit of T1 is $90 and per unit of T2 is $110. How many of each type of tables should be produced in order to maximize the total monthly profit?

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 $10 and contains 40 units of proteins, 20 units of minerals and 10 units of vitamins. A bag of food B costs $12 and contains 30 units of proteins, 20 units of minerals and 30 units of vitamins. How many bags of food A and B should the consumed by the animals each day in order to meet the minimum daily requirements of 150 units of proteins, 90 units of minerals and 60 units of vitamins at a minimum cost?

1. max profit = 90x₁ + 110x₂

Subject to:

x₁, x₂ 0

2x₁ + 4x₂ 7,000

x₁ + 2.5x₂ 4,000

2x₁ + 1.5x₂ 5,500

2. min cost = 10x₁ +12x₂

Subject to:

x₁ ,x₂ 0

40x₁ + 30x₂ 150

20x₁ + 20x₂ 90

10x₁ + 30x₂ 60