A diet is to include at least 140 milligrams of vitamin A and at least 140 milligrams of vitamin B. These requirements are to be obtained from two types of food. Type X contains 10 milligrams of vitamin A and 20 milligrams of vitamin B per pound. Type Y contains 30 milligrams of vitamin A and 50 milligrams of vitamin B per pound. If type X food costs $12.00 per pound and type Y food costs $8.00 per pound how many pounds of each type of food should be purchased to satisfy the requirements at the minimum costs? What is the minimum cost?

Step-by-step explanation:

The quantity of vitamin A in the food is ...

  10X +30Y ≥ 140

and vitamin B ...

  20X +50Y ≥ 140

The cost, which we want to minimize is ...

  12X +8Y

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A graph shows that any diet meeting the requirement for vitamin A will also meet the requirement for vitamin B. Vitamin A in the form of cereal X costs ...

  $12/(10 mg) = $1.20/mg

In the form of cereal Y, it costs ...

  $8/(30 mg) ≈ $0.27/mg

So, cost is minimized by satisfying the vitamin requirements using cereal Y exclusively. That cost is $37.33 for 4 2/3 pounds of cereal Y only.