Chapter 6.4, Problem 37E

In Problems 33-38, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model). Then solve the problem using the big $M$ method.

Plant food. A farmer can use three types of plant food: mix $A$, mix $B$, and mix $C$. The amounts (in pounds) of nitrogen, phosphoric acid, and potash in cubic yard of each mix are given in the table. Tests performed on the soil indicate that the field needs at least $800$ pounds of potash. The tests also indicate that no more than $700$ pounds of phosphoric acid should be added to the field. The farmer plans to plant a crop that requires a great deal of nitrogen. How many cubic yards of each mix should be added to the field in order to satisfy the potash and phosphoric acid requirements and maximize the amount of nitrogen? What is the maximum amount of nitrogen?