An electricity producer has two power plants, each of which emits carbon dioxide (CO₂), a greenhouse gas. Each plant is currently emitting 1 million metric tons of CO₂ per year. However, new emissions rules restrict the firm’s emissions to 1 million metric tons of CO₂ per year from both plants combined. The cost of operating a power plant goes up as it curtails its emissions. The following table shows the cost of operating each plant for different emissions levels:

The firm’s goal is to choose emissions levels at each plant that minimize its total cost of operating its plants, subject to meeting its emissions target of 1 million metric tons of CO₂ per year from both plants combined. Let X denote the quantity of emissions from plant 1 and Y denote the quantity of emissions from plant 2. Let TC(X, Y) denote the total operating cost of the firm when the quantity of emissions from plant 1 is X and the quantity of emissions from plant 2 is Y.
a) What is the objective function for this problem?
b) What is the constraint?
c) Write a statement of the constrained optimization problem.
d) In light of the information in the table, what emissions levels from each plant should the firm choose?

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Emissions of CO₂ by a Plant (metric tons per year) 0 250,000 500,000 750,000 1,000,000 Annual Cost of Operating Plant 1 (millions) $490 $360 $250 $160 $ 90 Annual Cost of Operating Plant 2 (millions) $250 $160 $ 90 $ 40 $ 10