Construct a mathematical model in the form of a linear programming problem. Then solve the problem using the big M method.

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 a 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?

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Nitrogen Phosphoric acid Potash Pounds per Cubic Yard A B C 12 16 12 8 16 8 8 16 16