Question

The cost \(C\) (in dollars) of removing \(p \%\) of the air pollutants in the stack emission of a utility company that burns coal is modeled by \(C=80,000 p /(100-p), \quad 0 \leq p<100\) (a) Find the costs of removing \(15 \%, 50 \%\), and \(90 \%\). (b) Find the limit of \(C\) as \(p \rightarrow 100^{-}\). Interpret the limit in the context of the problem. Use a graphing utility to verify your result.

Short answer

The costs of removing 15%, 50%, and 90% pollutants are $14,117.647, $80,000, and $720,000 respectively. As the percentage of pollutants removal approaches 100%, the cost increases without bound, meaning it becomes prohibitively expensive. This result is confirmed by the graph of the function.

Step-by-step solution

Finding the costs of removing 15%, 50%, and 90% pollutants.

Substitute \(p = 15, 50,\) and \(90\) into the cost function \(C=80,000 p /(100-p)\) and calculate: For \(p = 15\), \(C = 80,000 \times 15/(100-15) = 14,117.647 dollars\), For \(p = 50\), \(C = 80,000 \times 50/(100-50) = 80,000 dollars\), For \(p = 90\), \(C = 80,000 \times 90/(100-90) = 720,000 dollars\)

Calculating Limit as \(p \rightarrow 100^{-}\)

To calculate this limit means to understand what happens to C as p gets very close to 100 from the left. Technically, substitute \(p = 100\) into the function and we get: \(C = 80,000 \times 100/ (100-100) = 800,000/0\). This is undefined and can be inferred as the cost going to infinity.

Interpretation of the Limit

The fact that the cost goes to infinity as the company tries to remove close to 100% of the pollutants suggests that the cost becomes prohibitively expensive as the percentage of pollutants to be removed gets very close to 100. It is impractical for the company to try to remove all the pollutants from their emissions.

Graphical Verification

Use any available graphing utility. Plot the function \(C=80,000 p /(100-p)\) for \(0 \leq p<100\). Observe the function graph rising steeply towards infinity as \(p \rightarrow 100^{-}\), thus confirming the calculations in the previous steps.

Key concepts

Cost Function

Understanding the cost function is fundamental for students grappling with economic and operational concepts. A cost function, in this context, represents the cost, usually in dollars, of performing a certain action—in this case, removing air pollutants emitted by a coal-burning utility company. The function provided,

costly pollution removal efforts can have on a company's finances. By learning about such a function, students can better grasp the balancing act between environmental responsibility and economic viability.

As they apply the given formula \(C = \frac{80,000p}{100 - p}\)nse considering the impact of increasing percentages of pollutant removal.

Percentage Pollutants Removal

The percentage of pollutants removal plays a significant role not only in environmental science but also in practical business decision-making. In the provided exercise, students are required to analyze the cost implications of removing different percentages of pollutants, specifically 15%, 50%, and 90%. This exploration serves an essential purpose: it illustrates how the costs escalate with the ambition to achieve a cleaner emission.

For example, removing 15% of pollutants incurs a relatively moderate cost, useful in understanding how different levels of environmental commitment can influence a company's expenses.

This percentage-based approach to pollution control also highlights the effectiveness of various environmental policies and regulations, offering a quantifiable measure of their impact on both the environment and business operations.

Asymptotic Behavior

Asymptotic behavior in algebra refers to the way a function behaves as it approaches a particular line or point. In the context of cost functions related to pollutant removal, this behavior reveals the intensity of increasing costs as the goal of pollutant elimination approaches completeness.

With the function\(C = \frac{80,000p}{100 - p}\),s this is known as an asymptote. Here, as the denominator approaches zero, the function's value surges towards infinity, indicating that striving for 100% pollutant removal is theoretically infinite in cost. This mathematical concept helps students grasp why certain goals, like total pollutant eradication, might not be feasible in real-world scenarios due to escalating costs.

Graphical Interpretation

A graphical interpretation of a function brings mathematical predictions into visual reality, providing an intuitive grasp of abstract concepts. Using graphing utilities, students can visualize how the cost function \(C = \frac{80,000p}{100 - p}\)his graphical perspective reinforces the theoretical findings obtained through algebraic manipulations and encourages students to appreciate the relationship between visualization tools and complex mathematical functions.