Joyce and Marvin run a day care for preschoolers. They are trying to decide what to feed the children for lunches. They would like to keep their costs down, but also need to meet the nutritional requirements of the children. They have already decided to go with peanut butter and jelly sandwiches, and some combination of graham crackers, milk, and orange juice. The nutritional content of each food choice and its cost are given in the table below.

The nutritional requirements are as follows. Each child should receive between 400 and 600 calories. No more than 30 percent of the total calories should come from fat. Each child should consume at least 60 milligrams (mg) of vitamin C and 12 grams (g) of protein. Furthermore, for practical reasons, each child needs exactly 2 slices of bread (to make the sandwich), at least twice as much peanut butter as jelly, and at least 1 cup of liquid (milk and/or juice).
Joyce and Marvin would like to select the food choices for each child which minimize cost while meeting the above requirements.
(a) Formulate a linear programming model for this problem.
(b) Solve this model by the simplex method.

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Calories Total Vitamin C Protein Cost Food Item from Fat Calories(mg) Bread (1 slice 10 Peanut butter (1 tbsp) Strawberry jelly (1 tbsp) Graham cracker (1 cracker) Milk (1 cup) Juice (1 cup) 0 75 100 4 4 50 20 60 150 100 0 2 120 15 35 0