Chapter 11.8, Problem 20AYU

Farm Management A farmer has 70 acres of land available for planting either soybeans or wheat. The cost of preparing the soil, the workdays required, and the expected profit per acre planted for each type of crop are given in the following table.

The farmer cannot spend more than $\$1800$ in preparation costs and cannot use a total of more than 120 workdays. How many acres of each crop should be planted to maximize the profit? What is the maximum profit? What is the maximum profit if the farmer is willing to spend no more than $\$2400$ on preparation?

To determine

To solve: The given linear programming problem.

Answer to Problem 20AYU

Solution:

1. The maximum profit is 5520, when planting 24 acres of soybeans and 12 acres of wheat.

2. The maximum profit is 7200, when planting 40 acres of soybeans.

Explanation of Solution

Given:

Calculation:

Step 1:

Begin by assigning symbols for the two variables.

$x=$ Area for planting Soyabeans (in acre).

$y=$ Area for planting Wheat (in acre).

If $P$ is the profit, then

$P=180x+100y$

Step 2:

The goal is to maximize $P$ subject to certain constraints on $x$ and $y$. Because $x$ and $y$ represents the area, the only meaningful values of $x$ and $y$ are non-negative.

Therefore, $x\ge 0,y\ge 0$.

Preparation cost and workdays are given. From that, we get

$60x+30y\le 1800$

$3x+4y\le 120$

Therefore, the linear programming problem may be stated as

$\text{Maximize} P=180x+100y$

Subject to

$x\ge 0$

$y\ge 0$

$x+y\le 70$

$60x+30y\le 1800 => 2x+y\le 60$

$3x+4y\le 120$

Step 3:

The graph of the constraints is illustrated in the figure below.

The below table lists the corner points and the corresponding values of objective function.

Corner points are	Value of objective function
$\left(30,0\right)$	5400
$\left(0,30\right)$	3000
$\left(24,12\right)$	5520

From the table,

1. The maximum profit is 5520, when planting 24 acres of soybeans and 12 acres of wheat.

2. If the preparation cost of $\$2400$ is considered, the linear programming problem becomes,

Maximize $P=180x+100y$

Subject to

$x\ge 0$

$y\ge 0$

$x+y\le 70$

$60x+30y\le 2400 => 2x+y\le 80$

$3x+4y\le 120$

The graph of the constraints is illustrated in the figure below.

The below table lists the corner points and the corresponding values of objective function.

Corner points are	Value of objective function
$\left(40,0\right)$	7200
$\left(0,30\right)$	3000

The maximum profit is 7200, when planting 40 acres of soybeans.