A fruit grower can use two types of fertilizer in her orange grove, brand A or brand B. Brand A contains 4lbs of Nitrogen, 4lbs of Phosphoric Acid, and 2lbs of Chloride. Brand B contains 5lbs of Nitrogen, 4lbs of Phosphoric Acid, and 1lb of Chloride. Tests indicate that the grove needs at least 600lbs of phosphoric acid and at most 250lbs of chloride.

If the grower wants to maximize the amount of nitrogen added to the grove, how many bags of each mix should be used? How much nitrogen will be added?

If the grower wants to minimize the amount of nitrogen added to the grove, how many bags of each mix should be used? How much nitrogen will be added?

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Let x represent the number of bags of brand A, y the number of bags of brand B.

The objective function is C=4x+5y (This represents the amount of nitrogen provided by a particular mix of fertilizers -- 4lbs from each A and 5 lbs from each B.)

The constraints are :

`4x+4y>=600` or `x+y>=150`

`2x+y<=250`

and the natural constraints `x,y>=0` .

The constraints form a bounded feasible region so the objective function has both a maximum and a minimum.

The maximum/minimum must occur at the boundary points which are (0,150),(0,250) and (100,50)

C(0,150)=750

C(0,250)=1250

C(100,50)=650

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Under the given restraints the maximum nitrogen is 1250lbs from 250 bags of brand B.

The minimum nitrogen is 650lbs from 100 bags of A and 50 bags of B

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The graph of the feasible region:

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