Chapter 6.3, Problem 56E

In Problems 49-58, construct a mathematical model in the form of a linear programming problem. (the answers in the back of the book for these application problems indicate the model.) then solve the problem by applying the simplex method to the dual problem.

Plant food. A farmer can buy three types of plant food: mix $A$, mix $B$, and mix $C$. Each cubic yard of mix $A$ contains $20$ pounds of phosphoric acis, $10$ pounds of nitrogen, and $10$ pounds of potash. Each cubic yard of mix $B$ contains $10$ pounds of phosphoric acis, $10$ pounds of nitrogen, and $15$ pounds of potash. Each cubic yard of mix $C$ contains $20$ pounds of phosphoric acis, $20$ pounds of nitrogen, and $5$ pounds of potash. The minimum monthly requirements are $480$ pounds of phosphoric acid, $320$ pounds of nitrogen, and $225$ pounds of pottash. If mix $A$ costs $\text{\$}30$ per cubic yard, mix $B$ costs $\text{\$}36$ per cubic yard, and mix $C$ costs $\text{\$}39$ per cubic yard, how many cubic yards of each mix should the farmer blend to meet the minimum monthly requirements at a minimal cost? What is the minimum cost?