Farmer Jones must determine how many acres of com and wheat to plant this year. An acre of wheat yields 25 bushels of wheat and requires 10 hours of labor per week. An acre of corn yields 10 bushels of corn and requires 4 hours of labor per week. All wheat can be sold at $4 a bushel, and all com can be sold at $3 a bushel. Seven acres of land and 40 hours per week of labor are available. Government regulations requite that at least 30 bushels of corn be produced during the current year. Let x1 = number of acres of corn planted, and x2 = number of acres of wheat planted. Using these decision variables, formulate an LP whose solution will tell Farmer Jones how to maximize the total revenue from wheat and corn.

Z_max = 30x1 + 100x2

x1 + x2 ≤ 7

4x1 + 10x2 ≤ 40

x1 ≥ 3

x1 ≥ 0

x2 ≥ 0

Step-by-step explanation:

We are told that;

x1 = number of acres of corn planted

x2 = number of acres of wheat planted

We are told that an acre of wheat yields 25 bushels of wheat.

Also that a wheat is sold at $4 a bushel.

Thus, revenue gotten from number of acres of wheat planted = 4 × 25x2 = $100x2

Similarly, for corn;

We are told that an acre of corn yields 10 bushels of corn. Also that corn is sold at $3 a bushel.

Thus, revenue gotten from number of corn planted = 3 × 10x1 = $30x1

Therefore, the objective function will be written as;

Z_max = 30x1 + 100x2

The first constraint we are given is that 7 acres of land are available. Thus;

x1 + x2 ≤ 7

Second constraint is that 40 hours per week of labor are available, Thus;

4x1 + 10x2 ≤ 40

The third constraint is that Government regulations requite that at least 30 bushels of corn be produced during the current year. Thus;

10x1 ≥ 30

This gives; x1 ≥ 3

Now, the number of acres of corn and wheat each planted can't be negative. Thus;

x1 ≥ 0

x2 ≥ 0

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