Chapter 3.3, Problem 20PPS

To determine

To find: the coordinate of vertices of feasible regions for the given system of inequalities, $y\le x+4,y\ge x-4,y\le -x+10\text{ and }y\ge -x-10$ and find maximum and minimum values of the function $f\left(x,y\right)=-10x+9y$ for this feasible regions.

Answer to Problem 20PPS

  $52$ and $-43$ .

Explanation of Solution

Given

Given inequalities and function are.

  $\begin{array}{l}y\le x+4\\ y\ge x-4\\ y\le -x+10\\ y\ge -x-10\\ f\left(x,y\right)=-10x+9y\end{array}$ .

Calculation:

Plotting the graph for given inequalities.

Plotting $y\le x+4$ .

Draw graph of the line $y=x+4$ for the given coordinates below.

  $\begin{array}{cc}x& y\\ 0& 4\\ -4& 0\end{array}$

Shade the region below the line $y\le x+4$ .

Plotting $y\ge x-4$ .

Draw graph of the line $y=x-4$ for the given coordinates below.

  $\begin{array}{cc}x& y\\ 0& -4\\ 4& 0\end{array}$

Shade the region above the line as $y\ge x-4$ .

Plotting $y\le -x+10$ .

Draw graph of the line $y=-x+10$ for the given coordinates below.

  $\begin{array}{cc}x& y\\ 0& 10\\ 10& 0\end{array}$

Shade the region below the line as $y\le -x+10$ .

Plotting $y\ge -x-10$ .

Draw graph of the line $y=-x-10$ for the given coordinates below.

  $\begin{array}{cc}x& y\\ 0& -10\\ -10& 0\end{array}$

Shade the region above the line as $y\ge -x-10$ .

  

Coordinates of the vertices of feasible region are given in table below.

  $\begin{array}{ccc}\left(x,y\right)& -10x+9y& =f\left(x,y\right)\\ \left(2,8\right)& -10\left(2\right)+9\left(8\right)& 52\\ \left(7,3\right)& -10\left(7\right)+9\left(3\right)& -43\\ \left(-3,-7\right)& -10\left(-3\right)+9\left(-7\right)& -33\\ \left(-8,-2\right)& -10\left(-8\right)+9\left(-2\right)& -18\end{array}$

Maximum value is $52$ at $\left(2,8\right)$ and minimum value is $-43$ at $\left(7,3\right)$ .

So the answer is $52$ and $-43$ .