1. Air Pollution Reduction - Cost-Benefit Analysis

Coal has long been a reliable source of American energy, but it comes with tremendous costs because it is very dirty. When coal is burned it releases a number of airborne toxins and pollutants, including mercury, lead, sulfur dioxide, nitrogen oxides, particulates, and various other heavy metals which can have harmful environmental impacts in addition to CO₂.

The function below relates the cost C (in $1000) to remove x percent of the air pollutants for a hypothetical power company which burns coal to generate electricity. A function such as this is called a cost-benefit function because it relates a cost (the price of implementing practices to remove the pollution) and a benefit (the removal of the air pollutants from the energy-generating process).

1. Use the above function to show that the cost of removing 40% of the air pollutants would be $360,000. Then compute the cost (in dollars) for the company to remove 50%, 55%, 60%, 65%, 70%, 75%, 80%, 85%, 90%, and 95% of the air pollutants. Organize your results in a table and make a scatterplot of the points using Desmos. What trend do you observe?

2. What would be the increased cost (in dollars) for the company if they were to add processes that would increase the amount of pollutants removed from 50% to 60%, from 60% to 70%, from 70% to 80%, and from 80% to 90%? Comment on any trend that may be noticed)

3. According to the cost-benefit function, would it be possible for the company to remove 100% of the air pollutants? Explain why this does or does not make sense.

4. The domain of is restricted to . Explain why this makes sense in the context of this model.

5. If the company decides they can reasonably budget $2 million for pollution control, what percentage of air pollutants can be removed (to the nearest tenth of a percent)?

6. Sketch the graph of () using Desmos. What is the range of ? Explain why the range makes sense in the context of the problem.