A Utility Company Burns Coal To Generate Electricity. The Cost, $C(p$\], In Dollars, Of Removing $p$ Percent Of The Smokestack Pollutants Is Given By:$C(p) = \frac{85000p}{100-p}$a) Find The Cost To Remove $70\%$ Of

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A Utility Company's Coal-Burning Conundrum: Calculating the Cost of Pollution Removal

As the world grapples with the challenges of climate change and environmental degradation, the role of utility companies in generating electricity has come under increasing scrutiny. One of the primary concerns is the impact of coal-burning power plants on air quality, with smokestack pollutants posing a significant threat to public health and the environment. In this article, we will explore the cost of removing pollutants from the smokestack of a coal-burning utility company, using a mathematical model to calculate the expenses involved.

The cost, C(p)C(p), in dollars, of removing pp percent of the smokestack pollutants is given by the function:

C(p)=85000p100−pC(p) = \frac{85000p}{100-p}

This function represents the relationship between the percentage of pollutants removed and the corresponding cost. To understand the behavior of this function, we need to analyze its components.

The cost function consists of two main components: the numerator and the denominator. The numerator, 85000p85000p, represents the cost of removing pp percent of the pollutants, while the denominator, 100−p100-p, represents the percentage of pollutants remaining in the smokestack.

To find the cost of removing 70% of the pollutants, we need to substitute p=70p=70 into the cost function:

C(70)=85000(70)100−70C(70) = \frac{85000(70)}{100-70}

Simplifying the expression, we get:

C(70)=595000030C(70) = \frac{5950000}{30}

C(70)=198333.33C(70) = 198333.33

Therefore, the cost of removing 70% of the pollutants is approximately $198,333.33.

The result indicates that removing 70% of the pollutants would cost the utility company approximately $198,333.33. This is a significant expense, but it is essential to consider the long-term benefits of reducing pollution, including improved air quality and public health.

To understand the sensitivity of the cost function to changes in the percentage of pollutants removed, we can perform a sensitivity analysis. This involves analyzing how the cost changes in response to small changes in the percentage of pollutants removed.

To calculate the marginal cost, we need to find the derivative of the cost function with respect to the percentage of pollutants removed. Using the quotient rule, we get:

dCdp=(100−p)(85000)−(85000p)(−1)(100−p)2\frac{dC}{dp} = \frac{(100-p)(85000) - (85000p)(-1)}{(100-p)^2}

Simplifying the expression, we get:

dCdp=85000(100)(100−p)2\frac{dC}{dp} = \frac{85000(100)}{(100-p)^2}

dCdp=8500000(100−p)2\frac{dC}{dp} = \frac{8500000}{(100-p)^2}

This represents the marginal cost of removing an additional percentage of pollutants.

The marginal cost represents the additional cost of removing an additional percentage of pollutants. As the percentage of pollutants removed increases, the marginal cost decreases, indicating that the cost of removing additional pollutants becomes less expensive.

In conclusion, the cost of removing 70% of the pollutants from the smokestack of a coal-burning utility company is approximately $198,333.33. The marginal cost of removing additional pollutants decreases as the percentage of pollutants removed increases, indicating that the cost of reducing pollution becomes less expensive over time. This analysis provides valuable insights for utility companies and policymakers seeking to reduce pollution and improve air quality.

Based on the analysis, we recommend that utility companies consider the following strategies to reduce pollution:

  1. Invest in pollution-reducing technologies: Implementing technologies that can reduce pollution, such as scrubbers and electrostatic precipitators, can help reduce the cost of pollution removal.
  2. Increase the percentage of pollutants removed: Gradually increasing the percentage of pollutants removed can help reduce the marginal cost of pollution removal.
  3. Develop policies to support pollution reduction: Governments and regulatory agencies can develop policies to support pollution reduction, such as offering incentives for companies that invest in pollution-reducing technologies.

By implementing these strategies, utility companies can reduce pollution, improve air quality, and contribute to a healthier environment for future generations.
A Utility Company's Coal-Burning Conundrum: Calculating the Cost of Pollution Removal - Q&A

In our previous article, we explored the cost of removing pollutants from the smokestack of a coal-burning utility company, using a mathematical model to calculate the expenses involved. In this article, we will address some of the most frequently asked questions related to the cost of pollution removal and provide additional insights into the topic.

Q: What is the cost of removing 90% of the pollutants from the smokestack?

A: To find the cost of removing 90% of the pollutants, we need to substitute p=90p=90 into the cost function:

C(90)=85000(90)100−90C(90) = \frac{85000(90)}{100-90}

Simplifying the expression, we get:

C(90)=765000010C(90) = \frac{7650000}{10}

C(90)=765000C(90) = 765000

Therefore, the cost of removing 90% of the pollutants is approximately $765,000.

Q: How does the cost of pollution removal change as the percentage of pollutants removed increases?

A: To understand how the cost of pollution removal changes as the percentage of pollutants removed increases, we can analyze the marginal cost. The marginal cost represents the additional cost of removing an additional percentage of pollutants.

Using the marginal cost function, we get:

dCdp=8500000(100−p)2\frac{dC}{dp} = \frac{8500000}{(100-p)^2}

As the percentage of pollutants removed increases, the marginal cost decreases, indicating that the cost of removing additional pollutants becomes less expensive.

Q: What are some strategies that utility companies can use to reduce pollution and improve air quality?

A: Some strategies that utility companies can use to reduce pollution and improve air quality include:

  1. Investing in pollution-reducing technologies: Implementing technologies that can reduce pollution, such as scrubbers and electrostatic precipitators, can help reduce the cost of pollution removal.
  2. Increasing the percentage of pollutants removed: Gradually increasing the percentage of pollutants removed can help reduce the marginal cost of pollution removal.
  3. Developing policies to support pollution reduction: Governments and regulatory agencies can develop policies to support pollution reduction, such as offering incentives for companies that invest in pollution-reducing technologies.

Q: How can governments and regulatory agencies support pollution reduction efforts?

A: Governments and regulatory agencies can support pollution reduction efforts by:

  1. Developing policies to incentivize pollution reduction: Offering incentives, such as tax credits or grants, for companies that invest in pollution-reducing technologies can help encourage pollution reduction.
  2. Implementing regulations to reduce pollution: Establishing regulations that require companies to reduce pollution can help drive pollution reduction efforts.
  3. Providing funding for pollution reduction initiatives: Providing funding for initiatives that support pollution reduction, such as research and development of new pollution-reducing technologies, can help accelerate pollution reduction efforts.

Q: What are some of the benefits of reducing pollution from coal-burning power plants?

A: Some of the benefits of reducing pollution from coal-burning power plants include:

  1. Improved air quality: Reducing pollution from coal-burning power plants can help improve air quality, reducing the negative impacts of pollution on public health.
  2. Reduced greenhouse gas emissions: Reducing pollution from coal-burning power plants can also help reduce greenhouse gas emissions, contributing to efforts to mitigate climate change.
  3. Increased energy efficiency: Implementing pollution-reducing technologies can also help increase energy efficiency, reducing the amount of energy required to generate electricity.

In conclusion, the cost of removing pollutants from the smokestack of a coal-burning utility company is a complex issue that requires careful analysis. By understanding the cost function and the marginal cost, utility companies and policymakers can make informed decisions about how to reduce pollution and improve air quality. By implementing strategies such as investing in pollution-reducing technologies, increasing the percentage of pollutants removed, and developing policies to support pollution reduction, we can reduce pollution, improve air quality, and contribute to a healthier environment for future generations.