Chapter 2.7, Problem 21E

To determine

To find: the maximum number of each monitor the plant should make to maximize profits.

Answer to Problem 21E

4 small monitor and 5 large monitors

Explanation of Solution

Given information:

The table of manufacturing analysis is shown below.

Machine	Hours need to make		Total hours available
	Small monitor	Large monitor
A	1	2	16
B	1	1	9
C	1	4	21

The profit on each monitor is $40.

Calculation:

Let x and y be the small and large monitors respectively; and the maximum profit function be $f\left(x,y\right)$ .

From the given data, the set of inequalities and the function deduced are shown below.

  $\begin{array}{l}x+2y\le 16\\ x+y\le 9\\ x+4y\le 24\\ f\left(x,y\right)=40x+40y\end{array}$

The graph for the given inequalities is as follows:

  

The vertices of the polygon boundary are $\left(0,6\right),\left(4,5\right),$ and $\left(9,0\right)$ . Put these points in the $f\left(x,y\right)$ and check the maximum value as follows:

  $\begin{array}{c}f\left(0,6\right)=40\left(0\right)+40\left(6\right)\\ =240\\ f\left(4,5\right)=40\left(4\right)+40\left(5\right)\\ =360\\ f\left(9,0\right)=40\left(9\right)+40\left(0\right)\\ =360\end{array}$

The maximum value is at two points $\left(4,5\right)\text{ and }\left(9,0\right)$ . There are alternate optimal solutions. However, considering the company most likely wants to produce both size monitors, the logical choice is 4 small monitors and 5 large monitors.