Question

A farmer has to plant soybeans and cotton. He has $1200 to spend. Each acre of soybeans costs $200 to plant, and each acre of cotton costs $100 to plant. The farmer has, at most, 12 hours to complete the planting. Each acre of soybeans takes 1 hour to plant, and each acre of cotton takes 2 hours to plant. If the profit is $500 per acre of soybeans and $300 per acre of cotton.

The objective function is to maximize profit, where x represents acres of soybeans and y represents acres of cotton.
Constraints: 200x + 100y = 1200, x + 2y = 12, x = 0, y = 0.
The objective function is to minimize cost, where x represents acres of soybeans and y represents acres of cotton.
Constraints: 500x + 300y = 1200, x + 2y =12, x= 0, y= 0.

How many acres of each crop should be planted to maximize profit?

Answer 1

To maximize profit, the farmer should use the constraints of cost and time to form a system of inequalities and then apply linear programming techniques to find the optimal number of acres of soybeans and cotton to plant based on an objective function of maximizing profit.

Step-by-step explanation:

The problem at hand is to determine how many acres of soybeans and cotton the farmer should plant to maximize profit. With $1200 to spend, each acre of soybeans costs $200, and each acre of cotton costs $100. The farmer also has a time constraint of at most 12 hours, with soybeans taking 1 hour per acre and cotton taking 2 hours per acre to plant. The profits from soybeans and cotton are $500 and $300 per acre, respectively.

Firstly, we define the constraints based on cost and time:

• 200x + 100y ≤ 1200 (Cost constraint)
• x + 2y ≤ 12 (Time constraint)
• x ≥ 0 (Non-negativity of soybeans)
• y ≥ 0 (Non-negativity of cotton)

Using these constraints, we can form a system of inequalities to define the feasible region for planting. Through graphical representation or using linear programming techniques like the Simplex method or the corner point method, the farmer can determine the optimal number of acres for each crop that maximizes profit.

The objective function for this linear programming problem is:

Maximize Profit = 500x + 300y

This function will be maximized at one of the corners of the feasible region defined by the constraints. By calculating the profit at each corner point, the farmer can choose the combination that offers the highest possible profit within the given constraints.