Chapter 3.4, Problem 14PPS

To determine

To calculate: The coordinates of the vertices of the feasible region formed by graphing the given inequalities $-9\le x\le -3,-9\le y\le -5\text{ and }3y+12x\le -75$ . Also maximum and minimum values of the function $f\left(x,y\right)=20x+8y$ in this region.

Answer to Problem 14PPS

The graph of the given system of inequalities is,

  

The coordinates of the feasible region are $\left(-9,-5\right),\left(-5,-5\right),\left(-9,-9\right)\text{ and }\left(-4,-9\right)$ .

The maximum value of the function is $-140\text{ at }\left(-5,-5\right)$ and minimum value of the function is $-\text{252 at }\left(-9,-9\right)$ .

Explanation of Solution

Given information:

The inequalities $-9\le x\le -3,-9\le y\le -5\text{ and }3y+12x\le -75$ and the function $f\left(x,y\right)=20x+8y$ .

Formula used:

Linear programming is a technique to find the maximum and the minimum value of a given function over a given system of some inequalities, with each inequality representing a constraint. Graph the inequalities and obtain the vertices of the feasible region (solution set). Substitute the coordinates of the feasible region in the function and determine the maximum and the minimum value.

Calculation:

Consider the provided system of inequalities $-9\le x\le -3,-9\le y\le -5\text{ and }3y+12x\le -75$ .

Recall that linear programming is a technique to find the maximum and the minimum value of a given function over a given system of some inequalities, with each inequality representing a constraint. Graph the inequalities and obtain the vertices of the feasible region (solution set).

Graph the given inequalities as,

  

In the above graph, red region represents the inequality $-9\le x\le -3$ , blue region represents the inequality $3y+12x\le -75$ and green region represents $-9\le y\le -5$ .

The feasible region is the region common to all the inequalities which is represented as,

  

The coordinates of the vertices of the feasible region are $\left(-9,-5\right),\left(-5,-5\right),\left(-9,-9\right)\text{ and }\left(-4,-9\right)$ .

Now, to find the maximum and minimum value of the function, substitute the coordinates of the feasible region in the function and determine the maximum and the minimum value.

So, substitute $\left(-9,-5\right),\left(-5,-5\right),\left(-9,-9\right)\text{ and }\left(-4,-9\right)$ in the function $f\left(x,y\right)=20x+8y$ and evaluate the maximum and minimum value as,

  $\begin{array}{ccc}\left(x,y\right)& 20x+8y& f\left(x,y\right)\\ \left(-9,-5\right)& 20\left(-9\right)+8\left(-5\right)& -220\\ \left(-5,-5\right)& 20\left(-5\right)+8\left(-5\right)& -140\\ \left(-9,-9\right)& 20\left(-9\right)+8\left(-9\right)& -252\\ \left(-4,-9\right)& 20\left(-4\right)+8\left(-9\right)& -152\end{array}$

From the above table, it is observed that the maximum value of the function is $-140\text{ at }\left(-5,-5\right)$ and minimum value of the function is $-\text{252 at }\left(-9,-9\right)$ .

Thus, graph of the given system of inequalities is,

  

The coordinates of the feasible region are $\left(-9,-5\right),\left(-5,-5\right),\left(-9,-9\right)\text{ and }\left(-4,-9\right)$ and the maximum value of the function is $-140\text{ at }\left(-5,-5\right)$ and minimum value of the function is $-\text{252 at }\left(-9,-9\right)$ .