Chapter 3.10, Problem 3E

To determine

To Find: To find the maximum and minimum value of the given objective function.

Answer to Problem 3E

Maximum value is 121 and minimum value is 15.

Explanation of Solution

Given information: Given function $S=10x+3y$

Graph:

The given inequalities are $0\le x\le 10,y\ge 5,\text{and}y\le -0.3x+10$

  

Calculation:

From the graph it can be seen that the feasible region is bounded by the points $(0,10),(0,5),(10,7)\text{and}(10,5)$

Now, the given function is $S=10x+3y$

  $\begin{array}{l}\text{at}(0,10),\text{we}\text{get}\\ S=10\times 0+3\times 10\\ S=0+30\\ S=30\\ \text{Thus,}\text{the}\text{value}\text{of}S\text{at}(0,10)\text{is}30.\\ \text{Similarly,}\\ \text{at}(0,5),\text{we}\text{get}\\ S=10\times 0+3\times 5\\ S=0+15\\ S=15\\ \therefore \text{at}(0,5)\text{the}\text{value}\text{of}S=15\\ \text{also,}\text{at}(10,7),\text{we}\text{get}\\ S=10\times 10+3\times 7\\ S=100+21\\ S=121\\ \therefore \text{at}(10,7)\text{the}\text{value}\text{of}S=121\\ \text{and}\text{at}(10,5),\text{we}\text{get}\\ S=10\times 10+3\times 5\\ S=100+15\\ S=115\\ \therefore \text{at}(10,5)\text{the}\text{value}\text{of}S\text{is}115.\end{array}$

Hence, the maximum value is 121 attain at (10,7) and minimum value is 15 attain at (0,5).