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 be the number of acres of oats planted and y be the number of acres of corn planted.
The constraints are:
x + y ≤ 70 (total available acreage constraint)
3x + 2y ≤ 150 (total available hours constraint)
The objective function is the profit equation:
Profit = 500x + 450y
To plot the feasible region, we first graph the lines x + y = 70 and 3x + 2y = 150, then shade the region where x + y ≤ 70 and 3x + 2y ≤ 150.
To find the maximum profit, we need to evaluate the profit equation at the corners of the feasible region. The corners of the feasible region are found by solving the system of equations:
x + y = 70
3x + 2y = 150
Solving this system gives x = 30 and y = 40.
Plugging these values into the profit equation:
Profit = 500(30) + 450(40) = $25,500
Therefore, the maximum profit is $25,500.