Chapter 7, Problem 65RE

Solution

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

The minimum value is 106.

Given Information:

The system of inequality is,

  $\begin{array}{c}\text{Objective function}:f=7x+6y\\ \text{Constraints:}\left\{\begin{array}{c}7x+5y\ge 100\\ 2x+5y\ge 50\\ x\ge 0,y\ge 0\end{array}\right.\\ x\ge 0,y\ge 0\end{array}$

Calculation:

Consider the given system of inequality,

  $\begin{array}{c}\text{Objective function}:f=7x+6y\\ \text{Constraints:}\left\{\begin{array}{c}7x+5y\ge 100\\ 2x+5y\ge 50\\ x\ge 0,y\ge 0\end{array}\right.\\ x\ge 0,y\ge 0\end{array}$

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

  

The corner points are $\left(0,20\right),\left(10,6\right),\left(25,0\right)$ .

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

Substitute 0 for x and 20 for y in the objective function.

  $\begin{array}{c}f=7\left(0\right)+6\left(20\right)\\ =120\end{array}$

Substitute 10 for x and 6 for y in the objective function.

  $\begin{array}{c}f=7\left(10\right)+6\left(6\right)\\ =70+36\\ =106\end{array}$

The minimum value is 106.

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.

Therefore, the minimum value is 106.