I am trying to find what the maximum profit is?

Question

A farmer has 110 acres on which to plant oats or corn. Each acre of oats requires $18 capital and 2 hours of labor. Each acre of corn requires $36 capital and 6 hours of labor. Labor costs are $8 per hour. The farmer has $2100 available for capital and $2400 available for labor. If the revenue is $51 from each acre of oats and $127 from each acre of corn, what planting combination will produce the greatest total profit? (Profit here is revenue plus leftover capital and labor cash reserve.) What is the maximum profit?

Answer 1

Let x = acres of corn planted and y = acres of oats.  The revenue is:

 

      R = $127x + $51y

 

The profit is revenue minus costs. The costs will be capital costs plus labor costs:

 

C = $36x + $18y + $8*(8x + 2y) = $100x + $34y

 

The profit is revenue - costs:

 

P = 127x + 51x - 100x - 34y = 27x + 27y

 

The constraints are:

 

      x + y ≤ 110 acres

      x ≥ 0

      y ≥ 0

      $36x + $18y ≤ $2100     (Capital Costs)

      6x + 2y ≤ 300               (Labor hours)

 

1. Graph the inequalities
2. Identify the feasible region
3. Determine the coordinates of the vertices of the feasible region
4. Plug each set of coordinates into the Profit equation to determine which set yields the greatest profit.