Q. 17 - Precalculus Enhanced with Graphing Utilities

Question

Solve each linear programming problem.

Minimize $5x+2y$ subject to $\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\le 10\\ 2x+y\ge 10\\ x+2y\ge 10\end{array}\right.$

Step-by-Step Solution

Answer

The maximum value is $50$ at $(10,0)$

Step 1. Given information

The given system of linear inequalities is :

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\le 10\\ 2x+y\ge 10\\ x+2y\ge 10\end{array}\right.$

The goal of the task is to solve the problem of linear programming. We are looking for the maximum value that can occur if x and y are solutions of the system. We draw a graph of the system and read the corner points from the graph.

Step 2. Draw the graph of the system

Step 3. Make a table for the corner points

Corner point (x,y)	$z=5x+2y$
$(0,10)$	$z=20$
$(10,0)$	$z=50$
$(\frac{10}{3},\frac{10}{3})$	$z=23.3$