Chapter 7, Problem 87RE

To determine

Find the minimum and maximum values of the objective function

Answer to Problem 87RE

Minimum value of $\boxed{z=26.25}$

Explanation of Solution

Given information:

Sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function (if possible) and where they occur, subject to the indicated constraints.

  $\begin{array}{l}Objective\\ function\\ z=1.75x+2.25y\end{array}$  $\begin{array}{l}Constra\mathrm{int}s\\ x\ge 0\\ y\ge 0\\ 2x+y\ge 25\\ 3x+2y\ge 45\end{array}$

Calculation:

We have to find minimum and maximum values of the objective function.

  $\begin{array}{l}Objective\\ function\\ z=1.75x+2.25y\end{array}$

With following constraints.

  $\begin{array}{l}Constra\mathrm{int}s\\ x\ge 0\\ y\ge 0\\ 2x+y\ge 25\\ 3x+2y\ge 45\end{array}$

The graph is given as,

  

Vertices of the region formed by constraints the objective function has following values,

  $\begin{array}{l}At\left(0,25\right)\\ z=1.75x+2.25y\\ z=1.75\times 0+2.25\times 25\\ z=56.25\end{array}$

  $\begin{array}{l}At\left(5,15\right)\\ z=1.75x+2.25y\\ z=1.75\times 5+2.25\times 15\\ z=42.5\end{array}$

  $\begin{array}{l}At\left(15,0\right)\\ z=1.75x+2.25y\\ z=1.75\times 15+2.25\times 0\\ z=26.25\end{array}$

Hence, Minimum value of $\boxed{z=26.25}$ there is no maximum value of $z$ .