Exercise 27 Page 208

Introducing Variables

Let's start with introducing variables for the number of pairs of indoor and outdoor soccer shoes made.

Writing Inequalities

Let's use these variables to write inequalities representing the constraints given in the question.

Inequality Related to Step 1 of the Manufacturing Process

In Step 1, each pair of outdoor shoes requires 2 hours of work, and each pair of indoor shoes requires 1 hour of work. The work needed to make o pairs of outdoor and i pairs of indoor shoes should not exceed the available 40 hours for Step 1. 2o+i ≤ 40

Inequality Related to Step 2 of the Manufacturing Process

In Step 2, each pair of outdoor shoes requires 1 hour of work, and each pair of indoor shoes requires 3 hours of work. The work needed to make o pairs of outdoor and i pairs of indoor shoes should not exceed the available 60 hours for Step 2. o+3i ≤ 60

Other Inequalities

The number of outdoor and indoor shoes made is certainly not negative. o≥ 0 i≥ 0

Graphing the Inequalities

Let's graph the inequalities corresponding to the constraints.

Graphing 2o+i≤ 40

The graph of the linear inequality 2o+i≤ 40 is a half plane bounded by the line we get by replacing the inequality symbol with equality. 2o+i= 40 We can graph this line by finding and connecting the intercepts.

• If o=0, then i=40
• If i=0, then 2o=40, so o=20.

Let's plot the intercepts and draw the line.

To see which half pane to shade, we test a point. The manufacturer needs 0+0=0 hours in Step 1 to make 0 pairs of outdoor and 0 pairs of indoor shoes. This is smaller than 40 hours, so the point (0,0) is in the shaded region.

Graphing o+3i≤ 60

We can use similar steps to graph the inequality o+3i≤ 60. For this inequality, the boundary line crosses the axes at (0,20) and (60,0).

Graphing o≥ 0 and i≥ 0

The inequalities o≥ 0 and i≥ 0 together mean that we only consider points in the first quadrant.

Graphing the Feasible Region

Let's now copy all graphs together on the same coordinate plane.

The feasible region is the overlapping part of the four graphs. We will need the vertices of this quadrilateral, so let's highlight these on the graph.

Finding the Vertices

The maximum profit corresponds to one of the vertices, so let's find the coordinates. We can see that one of the vertices is the origin. While graphing the boundary lines, we already found the coordinates of the other vertices on the axes. (0,0) and (0,20) and (20,0) To find the fourth vertex, we need to find the intersection of the corresponding lines. 2o+i=40 & (I) o+3i=60 & (II) Rearranging equation (I) gives i=40-2o. Let's substitute this into Equation (II).

Let's substitute this into Equation (I) to find the value of i.

We can add this fourth point to our vertex list. (0,0) and (0,20) and (20,0) and (12,16)

Finding the Profit

We are given that the profit on a pair of outdoor shoes is $20, and the profit on a pair of indoor shoes is $15. We can use these values to get an expression for the profit on o pairs of outdoor and i pairs of indoor shoes. P=20o+15i

Maximizing the Profit

We know that the profit is maximized at one of the vertices of the feasible region we found above. Let's find the profit at vertex (12,16).

We can find the profit at the other vertices similarly.

We can see that the maximum daily profit the manufacturer can achieve is $480 when he makes and sells 12 pairs of outdoor and 16 pairs of indoor shoes.