Chapter 3.1, Problem 3P

Explanation of Solution

Reformulating Farmer Jones’s LP:

From the given scenario, it is observed that farmer Jones must determine the number of corn and wheat to plant this year.

Let us consider,

• $x_1$ is the variable to represent the number of bushels of corn produced.
• $x_2$ is the variable to represent the number of bushels of corn produced.

Wheat can be sold at $4 a bushel and corn can be sold at $3 a bushel.

The object of the farmer is to maximize the total revenue from wheat and corn.

The objective function is as follows:

$\text{Maximize}{\text{z = 3x}}_{\text{1}}{\text{+4x}}_{\text{2}}$

Considering the constraints,

Constraint 1: Seven acres of land is available.

Constraint 2: 40 hours of labor are available each week.

Constraint 3: At least 30 bushels of corn must be produced.

Expressing the constraint 1 in terms of $x_1$ and $x_2$:

$\left\{\begin{array}{l}\text{(1/10 acers of land produces 1 bushel of corn)}\times \text{(number of bushels of corn produced)+}\\ \text{(1/25 acers of land produces 1 bushel of wheat)}\times \text{(number of bushels of wheat produced)}\end{array}\right\}\le 7$

$\frac{x_1}{10}+\frac{x_2}{25}\le 7$

Expressing the constraint 2 in terms of $x_1$ and $x_2$:

$\begin{array}{l}\{\end{array}$