Cost-Benefit Analysis Of Pollutant Removal Model And Calculations

$$$$$$

Understanding the Cost-Benefit Model

At the heart of our analysis lies the cost-benefit model represented by the equation $f(x) = \frac{6.8x}{100-x}$. This equation mathematically expresses the relationship between the percentage of pollutant removed ($x$) and the cost associated with that removal ($f(x)$), measured in thousands of dollars. To truly grasp the model's significance, we need to break down its components and understand their interplay.

The variable $x$ represents the percentage of pollutant removed, ranging from 0% (no removal) to 100% (complete removal). As we increase $x$, we are essentially increasing the level of environmental cleanup we are aiming for. The function $f(x)$ then quantifies the cost associated with achieving that specific level of pollutant removal. The numerator, $6.8x$, suggests a direct relationship between the percentage of pollutant removed and the cost. As $x$ increases, the numerator also increases, indicating a higher cost. However, the denominator, $100-x$, introduces a critical element of diminishing returns. As $x$ approaches 100, the denominator approaches zero, causing the overall cost $f(x)$ to increase dramatically. This reflects the reality that removing the last few percentages of pollutants often becomes significantly more expensive than removing the initial percentages.

This model is a powerful tool for decision-makers because it provides a quantitative framework for evaluating the economic implications of environmental cleanup efforts. By plugging in different values of $x$, we can estimate the cost associated with various levels of pollutant removal. This allows for a more informed discussion about the trade-offs between cost and environmental impact. However, it's crucial to remember that this model is a simplification of a complex reality. It does not account for all the factors that might influence the cost of pollutant removal, such as the specific type of pollutant, the location of the pollution, and the available technology. Therefore, while the model provides valuable insights, it should be used in conjunction with other information and expert judgment.

(a) Finding the Cost of Removing Specific Percentages of Pollutants

The first step in applying our cost-benefit model is to determine the cost associated with removing specific percentages of pollutants. This involves substituting different values of $x$ into the equation $f(x) = \frac{6.8x}{100-x}$ and calculating the corresponding cost $f(x)$. Let's explore how this works in practice.

Suppose we want to find the cost of removing 25% of the pollutant. In this case, we would substitute $x = 25$ into the equation:

$f(25) = \frac{6.8 * 25}{100 - 25} = \frac{170}{75} ≈ 2.27$

This result tells us that it would cost approximately $2.27 thousand dollars (or $2,270) to remove 25% of the pollutant. Now, let's consider a scenario where we aim to remove 50% of the pollutant. Substituting $x = 50$ into the equation, we get:

$f(50) = \frac{6.8 * 50}{100 - 50} = \frac{340}{50} = 6.8$

This calculation reveals that removing 50% of the pollutant would cost $6.8 thousand dollars (or $6,800). Notice that the cost has more than doubled compared to removing 25% of the pollutant, highlighting the non-linear nature of the cost-benefit relationship.

To further illustrate this point, let's calculate the cost of removing 75% of the pollutant. Substituting $x = 75$ into the equation:

$f(75) = \frac{6.8 * 75}{100 - 75} = \frac{510}{25} = 20.4$

The cost of removing 75% of the pollutant jumps to $20.4 thousand dollars (or $20,400). This significant increase demonstrates the principle of diminishing returns – as we remove a higher percentage of pollutants, the cost of removing each additional percentage point increases substantially. This pattern is a crucial consideration for policymakers and environmental managers when setting targets and allocating resources for pollution control.

By calculating the cost for various percentages of pollutant removal, we gain a clearer understanding of the economic trade-offs involved. This information can then be used to inform decisions about the optimal level of pollution control, balancing the costs of removal with the benefits of a cleaner environment. However, it's essential to remember that this model is just one piece of the puzzle. Other factors, such as the severity of the pollution, the potential health impacts, and the availability of resources, should also be taken into account.

(b) Determining the Percentage of Pollutants Removable Within a Budget

While understanding the cost of removing specific percentages of pollutants is crucial, another critical question is: given a specific budget, what percentage of pollutants can we realistically remove? This involves reversing our approach and solving the equation for $x$ instead of $f(x)$. Let's explore this process and its implications.

Suppose we have a budget of $5 thousand dollars for pollutant removal. This means we need to find the value of $x$ that satisfies the equation $f(x) = 5$. Substituting this into our cost-benefit model, we get:

$5 = \frac{6.8x}{100-x}$

Now, we need to solve this equation for $x$. First, we can multiply both sides by $(100 - x)$ to get rid of the fraction:

$5(100 - x) = 6.8x$

Expanding the left side, we have:

$500 - 5x = 6.8x$

Next, we can add $5x$ to both sides to group the $x$ terms:

$500 = 11.8x$

Finally, we can divide both sides by 11.8 to isolate $x$:

$x = \frac{500}{11.8} ≈ 42.37$

This result indicates that with a budget of $5 thousand dollars, we can remove approximately 42.37% of the pollutant. This type of calculation is invaluable for resource allocation and budget planning. It allows decision-makers to determine the maximum achievable level of pollutant removal given their financial constraints.

Let's consider another scenario with a larger budget of $10 thousand dollars. Following the same steps, we would set $f(x) = 10$ and solve for $x$:

$10 = \frac{6.8x}{100-x}$

$10(100 - x) = 6.8x$

$1000 - 10x = 6.8x$

$1000 = 16.8x$

$x = \frac{1000}{16.8} ≈ 59.52$

With a budget of $10 thousand dollars, we can remove approximately 59.52% of the pollutant. Again, we observe the principle of diminishing returns. Doubling the budget from $5 thousand to $10 thousand dollars does not double the percentage of pollutant removed. This highlights the importance of carefully considering the cost-effectiveness of different pollution control strategies.

By understanding how to calculate the percentage of pollutants removable within a given budget, policymakers can make more informed decisions about resource allocation and environmental targets. This approach ensures that resources are used efficiently and effectively to achieve the desired level of environmental cleanup. However, it's crucial to remember that the cost-benefit model is just one tool in the decision-making process. Other factors, such as the environmental and health benefits of pollutant removal, should also be considered.

(c) Identifying the Limit to Percent Removable

Our cost-benefit model, $f(x) = \frac{6.8x}{100-x}$, reveals a crucial limitation in pollutant removal efforts: there's a theoretical limit to the percentage of pollutants that can be removed, regardless of the budget. This limit is not explicitly stated but is inherent in the structure of the equation. To understand this limitation, we need to examine the behavior of the function as $x$ approaches a certain value.

As we discussed earlier, the denominator of the function, $(100 - x)$, plays a crucial role in determining the cost. As $x$ (the percentage of pollutant removed) gets closer and closer to 100, the denominator approaches zero. This has a dramatic effect on the cost, $f(x)$. When we divide any number by a value that is getting progressively smaller and closer to zero, the result becomes increasingly large. In mathematical terms, we say that the function approaches infinity as $x$ approaches 100.

This means that the cost of removing pollutants skyrockets as we attempt to remove a higher and higher percentage, approaching 100%. In practical terms, this signifies that it becomes prohibitively expensive, and eventually impossible, to remove every last trace of the pollutant. There will always be a point where the cost of removing an additional fraction of a percent far outweighs the benefits gained.

To illustrate this, imagine trying to remove 99% of the pollutant. Substituting $x = 99$ into the equation, we get:

$f(99) = \frac{6.8 * 99}{100 - 99} = \frac{673.2}{1} = 673.2$

This calculation shows that removing 99% of the pollutant would cost $673.2 thousand dollars (or $673,200). Now, let's consider the cost of removing 99.9% of the pollutant. Substituting $x = 99.9$:

$f(99.9) = \frac{6.8 * 99.9}{100 - 99.9} = \frac{679.32}{0.1} = 6793.2$

The cost of removing 99.9% of the pollutant jumps to a staggering $6793.2 thousand dollars (or $6,793,200). This exponential increase in cost highlights the theoretical limit to pollutant removal. Even a small increase in the target percentage, from 99% to 99.9%, results in a tenfold increase in cost.

This concept has significant implications for environmental policy and decision-making. It suggests that there's an optimal level of pollutant removal, a point where the benefits of further cleanup are no longer worth the drastically increased cost. Policymakers need to carefully weigh the costs and benefits of different levels of pollution control and set realistic and achievable targets.

Understanding this limit also encourages the exploration of alternative strategies. Instead of focusing solely on removing pollutants after they have been released, preventative measures can be more cost-effective in the long run. This could include adopting cleaner technologies, reducing pollution at the source, and implementing stricter environmental regulations. By combining pollutant removal efforts with preventative strategies, we can achieve a more sustainable and cost-effective approach to environmental protection.

In conclusion, the cost-benefit model reveals a fundamental limitation to pollutant removal. As we approach 100% removal, the cost increases exponentially, making complete removal impractical. This understanding is crucial for informed decision-making in environmental policy and resource allocation, guiding us towards a balanced approach that considers both environmental goals and economic realities.

In summary, the cost-benefit model $f(x) = \frac{6.8x}{100-x}$ provides a valuable framework for analyzing the economic implications of pollutant removal. By understanding the relationship between the percentage of pollutant removed and the associated cost, we can make more informed decisions about environmental policy and resource allocation. We've explored how to calculate the cost of removing specific percentages of pollutants, determine the percentage removable within a budget, and identify the theoretical limit to pollutant removal. This knowledge empowers us to strike a balance between environmental goals and economic constraints, fostering a more sustainable and responsible approach to environmental protection. Remember, this model is a simplification, and real-world decisions require consideration of various factors beyond the model's scope. However, it serves as a powerful tool for understanding the core economic trade-offs involved in environmental cleanup efforts.