Problem 72 - Precalculus : Building Concepts and Connections

Question

The annual cost, in millions of dollars, of removing arsenic from drinking water in the United States can be modeled by the function $$ C(x)=\frac{1900}{x} $$ where \(x\) is the concentration of arsenic remaining in the water, in micrograms per liter. A microgram is \(10^{-6}\) gram. (Source: Environmental Protection Agency) (a) Evaluate \(C(10)\) and explain its significance. (b) Evaluate \(C(5)\) and explain its significance. (c) What happens to the cost function as \(x\) gets closer to zero?

Step-by-Step Solution

Answer

The cost for removing arsenic from the drinking water is 190 million dollars when the concentration of arsenic is 10 micrograms per liter and it's 380 million dollars when the concentration is 5 micrograms per liter. As the desired concentration of arsenic becomes very close to zero, the cost increases exponentially.

Step 1: Understanding the function

The function \(C(x)=\frac{1900}{x}\) represents the annual cost in millions of dollars to remove arsenic from drinking water, where \(x\) is the remaining concentration of arsenic.

Step 2: Evaluating \(C(10)\)

To find the annual cost when the concentration of arsenic is 10 micrograms per liter, substitute \(x=10\) into the function: \(C(10)=\frac{1900}{10}=190\). This means it will cost 190 million dollars per year.

Step 3: Evaluating \(C(5)\)

To calculate the annual cost when the arsenic concentration is 5 micrograms per liter, use the function with \(x=5\): \(C(5)=\frac{1900}{5}=380\). This means it will cost 380 million dollars per year.

Step 4: Cost as \(x\) approaches zero

As \(x\) gets closer to zero (meaning that the concentration of arsenic becomes negligible), the value of the function \(C(x)\) increases exponentially. This is due to the inverse relationship between \(C(x)\) and \(x\) in the function \(\frac{1900}{x}\). Therefore, the cost of removing arsenic from drinking water becomes extremely high as the desired concentration of arsenic approaches zero.

Key Concepts

Environmental Mathematics

Environmental mathematics is a field that utilizes mathematical tools and principles to tackle environmental issues, such as pollution, sustainability, and resource management. In our exercise, we apply environmental mathematics to model the cost of removing arsenic from drinking water. The function
\[ C(x)=\frac{1900}{x} \]
quantifies the financial impact of reducing arsenic concentrations. Such models are crucial because they enable policymakers and engineers to predict costs and optimize strategies for environmental remediation. When evaluating \(C(10)\), for instance, the $190 million figure informs us about the financial commitment needed to achieve a specific arsenic concentration level.

Environmental mathematics often involves real-world problem-solving. It’s not just about raw data but also the interpretation, which can directly affect public health and economic decisions. Furthermore, understanding the influence of human activities on the environment and how mathematics can help mitigate negative impacts is an integral part of environmental mathematics.

Inverse Functions

Inverse functions essentially 'undo' the operations of the original function. In our context, if the cost function \(C(x)\) tells us the cost for a certain arsenic concentration, an inverse function would tell us the arsenic concentration for a given cost.

The cost function \(C(x)=\frac{1900}{x}\) is an example of an inverse relationship. As the arsenic concentration \(x\) decreases, the cost \(C(x)\) increases. This relationship is characteristic of inverse functions—when one variable increases, the other decreases, assuming the other variables remain constant. In evaluating different \(x\) values, such as in \(C(10)\) and \(C(5)\), we see this inverse relationship manifest through the increasing costs. Understanding inverse functions is vital in interpreting such relationships and using them to predict outcomes or draft policies in environmental management.

Limiting Behavior Mathematics

Limiting behavior in mathematics describes what happens to a function as the input approaches a certain value, often at the boundaries of the function's domain. In terms of environmental economics and our arsenic removal cost function, the limiting behavior of \(C(x)\) as \(x\) approaches zero is of particular interest.

With \(C(x)\) defined as \(C(x)=\frac{1900}{x}\), as \(x\) gets closer to zero, \(x\) represents an increasingly lower concentration of arsenic, possibly approaching 'pure' water. The function suggests that the cost soars towards infinity—a conceptual standpoint demonstrating that achieving absolute zero concentration, or complete purity, is both practically impossible and economically unfeasible.

An understanding of limiting behavior helps us grasp the realities and trade-offs in environmental remediation efforts. It reveals that beyond a certain point, efforts to remove a pollutant may yield diminishing returns and become prohibitively expensive—highlighting the importance of establishing sensible regulatory standards that balance health, environment, and economic considerations.