Chapter 8.6, Problem 21PE

A furniture manufacturer builds tables. The cost for materials and labor to build a kitchen table is $\text{\$}240$ and the profit is $\$160.$ The cost to build a dining room table is $\text{\$}320$ and the profit is $\text{\$}240.$ (See Examples 2-3)

Let $x$ represent the number of kitchen tables produced per month. Let $y$ represent the number of dining room tables produced per month.

a. Write an objective function representing the monthly profit for producing and selling $x$ kitchen tables and $y$ dining room tables.

b. The manufacturing process is subject to the following constraints. Write a system of inequalities representing the constraints.

• The number of each type of table cannot be negative.

• Due to labor and equipment restrictions, the company can build at most $120$ kitchen tables.

• The company can build at most $90$ dining room tables.

• The company does not want to exceed a monthly cost of $\text{\$}48,000.$

c. Graph the system of inequalities represented by the constraints.

d. Find the vertices of the feasible region.

e. Test the objective function at each vertex.

f. How many kitchen tables and how many dining room tables should be produced to maximize profit? (Assume that all tables produced will be sold.)

g. What is the maximum profit?