Solve this problem using the simplex method. (Answers should be accurate to at least 2 decimal places.)

A farmer named Jane has a 320 acre farm on which she plants two crops: corn and soybeans. For each acre of corn planted, Jane's expenses are $80 and for each acre of soybeans planted, her expenses are $100. Each acre of corn requires 100 bushels of storage and yields a profit of $60; each acre of soybeans requires 40 bushels of storage and yields a profit of $90.

The total amount of storage space available is 19200 bushels and Jane has only $20000 on hand.
(a) How many acres of each crop should Jane plant in order to maximize her profit?
 acres of corn
 acres of soybeans

(b) What will Jane's profit be if she follows this strategy?

To solve this problem using the **simplex method**, we will first formulate it as a linear programming (LP) problem. Here's a step-by-step breakdown:

### 1. Decision Variables
Let:
- $ x_1 $ = number of acres of corn Jane plants.
- $ x_2 $ = number of acres of soybeans Jane plants.

### 2. Objective Function
We are asked to maximize Jane's profit. The profit per acre of corn is $60, and the profit per acre of soybeans is $90. Thus, the objective function is:

\[
\text{Maximize } Z = 60x_1 + 90x_2
\]

### 3. Constraints
We now need to consider the constraints:

1. **Storage Constraint**: Each acre of corn requires 100 bushels of storage, and each acre of soybeans requires 40 bushels of storage. Jane has 19,200 bushels of storage available:

\[
100x_1 + 40x_2 \leq 19200
\]

2. **Budget Constraint**: Each acre of corn costs $80, and each acre of soybeans costs $100. Jane has $20,000 to spend:

\[
80x_1 + 100x_2 \leq 20000
\]

3. **Land Constraint**: Jane has 320 acres of land available:

\[
x_1 + x_2 \leq 320
\]

4. **Non-negativity Constraint**: She cannot plant a negative number of acres, so:

\[
x_1 \geq 0 \quad \text{and} \quad x_2 \geq 0
\]

### 4. Formulating the LP Problem
The linear programming problem is:

\[
\text{Maximize } Z = 60x_1 + 90x_2
\]
subject to:
\[
100x_1 + 40x_2 \leq 19200 \quad \text{(Storage constraint)}
\]
\[
80x_1 + 100x_2 \leq 20000 \quad \text{(Budget constraint)}
\]
\[
x_1 + x_2 \leq 320 \quad \text{(Land constraint)}
\]
\[
x_1 \geq 0, \, x_2 \geq 0
\]

### 5. Setting up for the Simplex Method
We convert the inequalities to equalities by introducing slack variables:
- Let $ s_1 $ be the slack variable for the storage constraint.
- Let $ s_2 $ be the slack variable for the budget constraint.
- Let $ s_3 $ be the slack variable for the land constraint.

This gives us the following system:

\[
100x_1 + 40x_2 + s_1 = 19200
\]
\[
80x_1 + 100x_2 + s_2 = 20000
\]
\[
x_1 + x_2 + s_3 = 320
\]

The objective function is:

\[
Z = 60x_1 + 90x_2
\]

### 6. Simplex Tableaux Setup
We can now set up the initial simplex tableau, introduce the basic variables, and perform the simplex iterations to find the optimal solution.

#### First Tableau:

\[
\begin{array}{c|ccccccc|c}
\text{BV} & x_1 & x_2 & s_1 & s_2 & s_3 & \text{Solution} \\
\hline
s_1 & 100 & 40  & 1  & 0  & 0  & 19200 \\
s_2 & 80  & 100 & 0  & 1  & 0  & 20000 \\
s_3 & 1   & 1   & 0  & 0  & 1  & 320   \\
\hline
-Z  & -60 & -90 & 0  & 0  & 0  & 0     \\
\end{array}
\]

Now, let's perform the simplex iterations to find the optimal solution. I'll calculate the result.

### Solution:

(a) To maximize profit, Jane should plant:
- $ \mathbf{0} $ acres of corn.
- $ \mathbf{200} $ acres of soybeans.

(b) Jane's maximum profit will be $ \mathbf{18,000} $ dollars.

Let me know if you'd like to see more details on the simplex iterations or clarifications!

### Additional Questions:
1. How does the budget constraint affect the number of acres Jane can plant?
2. What happens if the storage space is increased by 5000 bushels?
3. How would the solution change if Jane could spend an additional $5000?
4. Can Jane maximize her profit by planting only corn instead of soybeans?
5. How would a decrease in profit per acre of soybeans affect the optimal solution?

### Tip:
When using the simplex method, always check if any constraints can be binding at the optimal solution to determine whether they limit the potential profit.

A farmer named Jane has a 320 acre farm on which she plants two crops: corn and soybeans. For each acre of corn planted, Jane's expenses are $80 and for each acre of soybeans planted, her expenses are $100. Each acre of corn requires 100 bushels of storage and yields a profit of $60; each acre of soybeans requires 40 bushels of storage and yields a profit of $90. The total amount of storage space available is 19200 bushels and Jane has only $20000 on hand. How many acres of each crop should Jane plant in order to maximize her profit? What will Jane's profit be if she follows this strategy?

Discover how to maximize profit by planting corn and soybeans on a 320-acre farm using the simplex method. This linear programming problem considers expenses, storage, and land constraints to find the optimal solution. Learn the step-by-step approach to solving real-world optimization problems in operations research.

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