Chapter 3.10, Problem 5E

To determine

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

Answer to Problem 5E

12 and 24 packages are to be sold.

Explanation of Solution

Given information: Given mix plan of nutrition.

Graph:

Let the number of the trailblazer mix package sold be x and the number of frontier mix package sold be y. Now, according to given information we have to maximize incomewith the given availability of corn and cereal.

Thus, we have to

Maximize $I=9.75x+9.50y$ which shows the income function that have to be maximize and

  $x+2y\le 60,\text{and}4x+3y\le 120$ which shows the availability of corn and cereals.

  

Calculation:

From the graph it can be seen that the feasible region is bounded by the points $(0,0),(12,24),(0,30)\text{and}(30,0)$

Now, the given function is $I=9.75x+9.50y$

  $\begin{array}{l}\text{at}(0,0),\text{we}\text{get}\\ I=9.75\times 0+9.50\times 0\\ I=0+0\\ I=0\\ \text{Thus,}\text{the}\text{value}\text{of}I\text{at}(0,0)\text{is}0.\\ \text{Similarly,}\\ \text{at}(12,24),\text{we}\text{get}\\ I=9.75\times 12+9.50\times 24\\ I=117+228\\ I=345\\ \therefore \text{at}(12,24)\text{the}\text{value}\text{of}I=345\\ \text{also,}\text{at}(0,30),\text{we}\text{get}\\ I=9.75\times 0+9.50\times 30\\ I=0+285\\ I=285\\ \therefore \text{at}(0,30)\text{the}\text{value}\text{of}I=285.\\ \text{and}\text{at}(30,0),\text{we}\text{get}\\ I=9.75\times 30+9.50\times 0\\ I=292.5+0\\ I=292.5\\ \therefore \text{at}(30,0)\text{the}\text{value}\text{of}I\text{is}\mathrm{292.5.}\end{array}$

Hence, the income is maximum $345 attain when the trailblazer mix sold 12 packages and the frontier mix sold 24 packages.