A local farmer knows that he is able to sell his oats for $500.00 per acre and his corn for $450.00 per acre. He has a total of 150 hours to work in the fields. He has 70 acres available to plant. He calculates that it will take 3 hours of labour per acre of oats and 2 hours of labour per acre of corn to plant and care for the crops.
Write the system of linear inequalities to represent this situation.
Sketch the feasible region.
Write the optimization equation.
Determine the maximum profit. Show all work.
Let x represent the number of acres of oats and y represent the number of acres of corn.
The constraints are:
1. The farmer has a total of 150 hours to work in the fields: 3x + 2y ≤ 150.
2. The farmer has 70 acres available to plant: x + y ≤ 70.
3. The number of acres cannot be negative: x ≥ 0, y ≥ 0.
The optimization equation is:
Profit = 500x + 450y.
To determine the maximum profit, we need to find the corner points of the feasible region and evaluate the profit function at each point.
To find the corner points, we solve the system of equations formed by the two constraints:
1. x + y = 70
2. 3x + 2y = 150
Solving this system, we find x = 50 and y = 20.
The corner points are (0, 0), (50, 0), (0, 70), and (50, 20).
Evaluating the profit function at each corner point:
Profit at (0, 0) = 500(0) + 450(0) = $0.
Profit at (50, 0) = 500(50) + 450(0) = $25,000.
Profit at (0, 70) = 500(0) + 450(70) = $31,500.
Profit at (50, 20) = 500(50) + 450(20) = $38,000.
The maximum profit is $38,000 when the farmer plants 50 acres of oats and 20 acres of corn.