Mathematics

Breathtakers, a health and fitness center, operates a morning fitness program for senior citizens. The program includes aerobic exercise, either swimming or step exercise, followed by a healthy breakfast in the dining room. Breathtakers’ dietitian wants to develop a breakfast that will be high in calories, calcium, protein, and fiber, which are especially important to senior citizens, but low in fat and cholesterol. She also wants to minimize cost. She has selected the following possible food items, whose individual nutrient contributions and cost from which to develop a standard breakfast menu are shown in the following table:

Breakfast Food

Calories

Fat
(g)

Cholesterol
(mg)

Iron
(mg)

Calcium
(mg)

Protein
(mg)

Fiber
(g)

Cost
($)

1

Bran Cereal (cup)

90

0

0

6

20

3

5

0.18

2

Dry Cereal (cup)

110

2

0

4

48

4

2

0.22

3

Oatmeal (cup)

100

2

0

2

12

5

3

0.10

4

Oat Bran (cup)

90

2

0

3

8

6

4

0.12

5

Egg

75

5

270

1

30

7

0

0.10

6

Bacon

35

3

8

0

0

2

0

0.09

7

Orange

65

0

0

1

52

1

1

0.40

8

Milk – 2% (cup)

100

4

12

0

250

9

0

0.16

9

Orange Juice (cup)

120

0

0

0

3

1

0

0.50

10

Wheat toast (slice)

65

1

0

1

26

3

3

0.07

The dietitian wants the breakfast to include at least 420 calories, 5 milligrams of iron, 400 milligrams of calcium, 20 grams of protein, and 12 grams of fiber. Furthermore, she wants to limit fat to no more than 20 grams and cholesterol to 30 milligrams.

The problem to solve:

In the diet example in this chapter, what would be the effect on the optimal solution of increasing the minimum calorie requirement for the breakfast to 500 calories? To 600 calories?

Increase the breakfast calorie requirement to 700 calories and reformulate the model to establish upper limits on the servings of each food item to what you think would be realistic and appetizing. Determine the solution for this reformulated model.

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