Chapter 6.2, Problem 47E

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.

Home construction. A contractor is planning a new housing development consisting of colonial, split-level, and ranch-style houses. A colonial house requires $\frac{1}{2}$ acre of land, $\text{\$}60,000$ capital, and $4,000$ labor-hours to construct, and returns a profit of $\text{\$}20,000$. A split-level house requires $\frac{1}{2}$ acre of land, $\text{\$}60,000$ capital, and $3,000$ labor-hours to construct, and returns a profit of $\text{\$}18,000$. A ranch house requires $1$ acre of land, $\text{\$}80,000$ capital, and $4,000$ labor-hours to construct, and returns a profit of $\text{\$}24,000$. The contractor has $30$ acres of land, $\text{\$}3,200,000$ capital, and $180,000$ labor-hours available. How many houses of each type should be constructed to maximize the contractor’s profit? What is the maximum profit?