Chapter 6.2, Problem 53E

In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution.

Animal nutrition. The natural diet of a certain animal consists of three foods: $A,B,$ and $C$. The number of units of calcium, iron, and protein in $1$ gram of each food and the average daily intake are given in the table. A scientist wants to investigate the effect of increasing the protein in the animal’s diet while not allowing the units of calcium and iron to exceed its average daily intakes. How many grams of each food should be used to maximize the amount of protein in the diet? What is the maximum amount of protein?

$\begin{array}{l}\text{ Units per Gram}\\ \begin{array}{ccccc}& \text{Food }A& \text{Food }B& \text{Food }C& \begin{array}{l}\text{Average Daily }\\ \text{ Intake}\left(\text{units}\right)\end{array}\\ \text{Calcium}& 1& 3& 2& 30\\ \text{Iron}& 2& 1& 2& 24\\ \text{Protein}& 3& 4& 5& 60\end{array}\end{array}$