Chapter 7, Problem 67RE

Solution

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

The minimum and maximum value is 205 and 292.

Given Information:

The system of inequality is,

  $\begin{array}{c}\text{Objective function}:f=3x+7y\\ \text{Constraints:}\left\{\begin{array}{c}5x+2y\ge 100\\ x+4y\ge 110\\ 5x+11y\le 460\\ x\ge 0,y\ge 0\end{array}\right.\end{array}$

Calculation:

Consider the given system of inequality,

  $\begin{array}{c}\text{Objective function}:f=3x+7y\\ \text{Constraints:}\left\{\begin{array}{c}5x+2y\ge 100\\ x+4y\ge 110\\ 5x+11y\le 460\\ x\ge 0,y\ge 0\end{array}\right.\end{array}$

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

  

The corner points are $\left(4,40\right),\left(10,25\right),\left(70,10\right)$ .

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

Substitute 4 for x and 40 for y in the objective function.

  $\begin{array}{c}f=3\left(4\right)+7\left(40\right)\\ =12+280\\ =292\end{array}$

Substitute 10 for x and 25 for y in the objective function.

  $\begin{array}{c}f=3\left(10\right)+7\left(25\right)\\ =30+175\\ =205\end{array}$

Substitute 70 for x and 10 for y in the objective function.

  $\begin{array}{c}f=3\left(70\right)+7\left(10\right)\\ =210+70\\ =280\end{array}$

Since the feasible region is bounded, this conclude that there is maximum value for the objective function.

The minimum value is 205 and the maximum value is 292.

Therefore, the minimum and maximum value is 205 and 292.