Textbook Problem

An oil company must determine how many barrels of oil to extract during each of the next two years. If the company extracts x₁ million barrels during year 1, each barrel can be sold for 80 − x₁ dollars. If the company extracts x₂ million barrels during year 2, each barrel can be sold for 85 − x₂ dollars. The cost of extracting x₁ million barrels during year 1 is 2x₁² million dollars, and the cost of extracting x₂ million barrels during year 2 is 3x₂² million dollars. A total of 20 million barrels of oil are available, and at most $250 million can be spent on extraction. Determine how the company can maximize its profit (revenues less costs) for the next two years.