Q. 8 - Precalculus Enhanced with Graphing Utilities

Question

Find the maximum and minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of the feasible points. 

$z=7x+5y$

Step-by-Step Solution

Answer

The minimum value is 15 and the maximum value is 65.

Step 1. Given information

The goal of the task is to find the maximum and minimum values of the function. To solve this task, we use the points given in the task
Function given is $z=7x+5y$

Points are : $(0,6),(5,6),(0,3),(5,2),(4,0)$

Step 2. Substitute (0,6) into the equation

$$\begin{gathered} z=7x+5y \\ z=7\left(0\right)+5\left(6\right) \\ z=30 \end{gathered}$$

Step 3. Substitute (5,6) into the equation

$$\begin{gathered} z=7x+5y \\ z=7\left(5\right)+5\left(6\right) \\ z=65 \end{gathered}$$

Step 4. Substitute (0,3) into the equation

$$\begin{gathered} z=7x+5y \\ z=7\left(0\right)+5\left(3\right) \\ z=15 \end{gathered}$$

Step 5. Substitute (5,2) into the equation

$$\begin{gathered} z=7x+5y \\ z=7\left(5\right)+5\left(2\right) \\ z=45 \end{gathered}$$

Step 6. Substitute (4,0) into the equation

$$\begin{gathered} z=7x+5y \\ z=7\left(4\right)+5\left(0\right) \\ z=28 \end{gathered}$$