A furniture company produces chairs and sofas. The chairs require 30 ft of wood, 7 Ib of foam rubber, and 6 sq yd of fabric. The sofas require 150 ft of wood, 70 lb of foam rubber, and 30 sq yd of fabric The company has 2250 ft of wood, 630 Ib of foam rubber and 390 sq yd of fabric The chairs can be sold for 500 each and the sofas for 1300 each. How many of each should be produced in order to maximize income? The fumilure company should produce square chairs and square sofas to maximize the income. (Type whole numbers.)

## Step 1: Defining Variables and Setting up the Objective Function<br />### Let $x$ be the number of chairs produced and $y$ be the number of sofas produced. The objective is to maximize income, which is given by the function $I = 500x + 1300y$.<br /><br />## Step 2: Setting up the Constraints<br />### The production is limited by the available resources. Wood constraint: $30x + 150y \le 2250$. Foam rubber constraint: $7x + 70y \le 630$. Fabric constraint: $6x + 30y \le 390$. Also, $x \ge 0$ and $y \ge 0$ since we cannot produce negative quantities.<br /><br />## Step 3: Simplifying the Constraints<br />### Divide the wood constraint by 30: $x + 5y \le 75$. Divide the foam rubber constraint by 7: $x + 10y \le 90$. Divide the fabric constraint by 6: $x + 5y \le 65$.<br /><br />## Step 4: Graphing the Constraints and Finding the Feasible Region<br />### Plot the lines $x + 5y = 75$, $x + 10y = 90$, $x + 5y = 65$, $x=0$, and $y=0$. The feasible region is the intersection of all these inequalities. The vertices of the feasible region are $(0,0)$, $(65,0)$, $(25,10)$, $(0,9)$, and $(0,6)$.<br /><br />## Step 5: Evaluating the Objective Function at Each Vertex<br />### $I(0,0) = 500(0) + 1300(0) = 0$. $I(65,0) = 500(65) + 1300(0) = 32500$. $I(25,10) = 500(25) + 1300(10) = 12500 + 13000 = 25500$. $I(0,9) = 500(0) + 1300(9) = 11700$. $I(0,6) = 500(0) + 1300(6) = 7800$.<br /><br />## Step 6: Determining the Maximum Income<br />### The maximum income is $32500, which occurs when $x=65$ and $y=0$.