Chapter 6.2, Problem 37E

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.

Manufacturing: resource allocation. A small company manufactures three different electronic components for computers. Component $A$ requires $2$ hours of fabrication and $1$ hour of assembly; component $B$ requires $3$ hours of fabrication and $1$ hour of assembly; and component $C$ requires $2$ hours of fabrication and $2$ hours of assembly. The company has up to $1,000$ labor-hours of fabrication time and $800$ labor-hours of assembly time available per week. The profit on each component, $A,B,$ and $C,$ is $\text{\$}7,\text{ \$}8,$ and $\text{\$}10,$ respectively. How many components of each type should the company manufacture each week in order to maximize the profit (assuming that all components manufactured can be sold)? What is the maximum profit?