An electricity producer has two power plants, each of which emits carbon dioxide (CO2), a greenhouse gas. Each plant is currently emitting 1,000,000 metric tons of CO2 per year. However, new emissions rules restrict the firm's emissions to 1,000,000 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 table below 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 1,000,000 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 Annual cost of operating Annual cost of operating (metric tons per year) plant 1 (millions) plant 2 (millions) 250,000 500,000 750,000 1,000,000 $490 S360 $250 $160 S90 S250 $160 S90 S40 S10