Chapter 7, Problem 66RE

Solution

To calculate: The minimum and minimum value, checked they exists, of the objective function $f$ .

The minimum value is 138.

Given Information:

The system of inequality is,

  $\begin{array}{c}\text{Objective function}:f=11x+5y\\ \text{Constraints:}\left\{\begin{array}{c}5x+2y\ge 60\\ 5x+8y\ge 120\\ x\ge 0,y\ge 0\end{array}\right.\end{array}$

Calculation:

Consider the given system of inequality,

  $\begin{array}{c}\text{Objective function}:f=11x+5y\\ \text{Constraints:}\left\{\begin{array}{c}5x+2y\ge 60\\ 5x+8y\ge 120\\ x\ge 0,y\ge 0\end{array}\right.\end{array}$

Use the graphing calculator to draw the inequality and find the corner points.

  

The corner points are $\left(0,30\right),\left(8,10\right),\left(24,0\right)$ .

Now, find the value of the objective function on corner points.

Substitute 0 for x and 30 for y in the objective function.

  $\begin{array}{c}f=11\left(0\right)+5\left(30\right)\\ =150\end{array}$

Substitute 24 for x and 0 for y in the objective function.

  $\begin{array}{c}f=11\left(8\right)+5\left(10\right)\\ =88+50\\ =138\end{array}$

Substitute 8 for x and 10 for y in the objective function.

  $\begin{array}{c}f=11\left(24\right)+5\left(0\right)\\ =264\end{array}$

Since the feasible region is unbounded and has  $x\ge 0$  and  $y\ge 0$ , and conclude that there is no maximum value for the objective function.

The minimum value is 138.

Therefore, the minimum value is 138.