a fruit grower can use two types of fertilizer in his orange grove, brand A and

Question

a fruit grower can use two types of fertilizer in his orange grove, brand A and B. the amounts (in pounds) of nitrogen, phosphoric acid, and chloride in a bag of each brand are given in the table. tests indicate that the grove needs at least 2000 pounds of phosphoric acid and ar most 450 pounds of chloride. Nitrogen (A) 6 (B) 5 Phosphoric acid (A) 5 (B) 5 Chloride (A) 2 (B) 1

1. If the grower wants to maximize the amount of nitrogen added to the grove, how many bags of each should be used? How much nitrogen will be added? The grower should add _____ bags of brand A and ____ bags of brand B to maximize the amount of nitogren added.

2. If the grower wants to maximize the amount of nitrogen added to the grove, how many bags of each should be used? How much nitrogen will be added? The grower should add _____ bags of brand A and ____ bags of brand B to minimize the amount of nitogren added The minimum amount of nitrogen that will be added is ____ pounds

Explanation / Answer

1) Let the number of A bags needed to maximize the amunt of nitrogen be 'a' and that of B be 'b'. Since the amount of phosphoric acid should be greater than 2000 pounds, we have 5a+5b >=2000, so a+b>=400

Since the amount of chloride should be less than 450 pounds, we have 2a+b <=450, since a+b>=400,

we get that a<=50, b>=350.

The amount of nitrogen would be 6a+5b. For maximizing the amount of nitrogen, the number of bags used should be maximum, so a+b = 450 then we get a=0, b=450. So, the grower should add 0 bags of A and 450 bags of B to maximize the amount of nitrogen added and the maximum amount of nitrogen added is 2250 pounds

2) For minimizing the amount of nitrogen, the number of bags used should be minimum, so a+b = 400. Now since a+b=400, is fixed, the bag which has minimum amount of nitrogen should be higher i.e., B bags should be higher. Then we get the solution as a=0, b=400. So, the grower should add 0 bags of A and 400 bags of B to minimize the amount of nitrogen added and the minimum amount of nitrogen added is 2000 pounds

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