Problem 61 - Single Variable Calculus

Question

Chemical Pollution As a result of an abandoned chemical dump leaching chemicals into the water, the main well of a town has been contaminated with trichloroethylene, a cancer-causing chemical. A proposal submitted by the town's board of health indicates that the cost, measured in millions of dollars, of removing \(x\) percent of the toxic pollutant is given by $$ C(x)=\frac{0.5 x}{100-x} $$ a. Evaluate \(\lim _{x \rightarrow 100-} C(x)\), and interpret your results. b. Plot the graph of \(C\) using the viewing window \([0,100] \times[0,10] .\)

Step-by-Step Solution

Answer

a. The limit of the cost function as \(x\) approaches 100 from the left is \(\lim _{x \rightarrow 100^-} C(x) = \frac{99}{2}\). This represents the theoretical cost in millions of dollars required to remove 100% trichloroethylene from the well. Practically, this cost might not be feasible as it increases substantially when \(x\) gets closer to 100. b. To plot the graph of \(C(x) = \frac{0.5 x}{100-x}\) within the specified viewing window \([0,100]\times[0,10]\), use graphing calculator programs or websites, like Desmos or GeoGebra. Observe how the graph rapidly increases as it approaches \(x=100\).

Step 1: Compute the Limit of C(x) as x Approaches 100

First, let's find the limit of the function \(C(x)=\frac{0.5 x}{100-x}\) as \(x\) approaches 100 from the left. The left-hand limit is denoted by \(\lim _{x \rightarrow 100^-} C(x)\). This can be found by direct substitution, but since the function is rational and undefined as x approaches 100, we have to substitute the value \(x = 100 - h\) as \(h\) approaches 0, where \(h\) is any positive number. \[ \lim _{h \rightarrow 0^+} C(100 - h) = \lim _{h \rightarrow 0^+} \frac{0.5 (100 - h)}{100 - (100 - h)} \]

Step 2: Simplify and Solve the Limit

Now let's simplify and solve the limit found in step 1: \[ \lim _{h \rightarrow 0^+} C(100 - h) = \lim _{h \rightarrow 0^+} \frac{50 - 0.5h}{h} \] Canceling h in numerator and denominator: \[ \lim _{h \rightarrow 0^+} C(100 - h) = \lim _{h \rightarrow 0^+} \frac{100 - 1}{2} \] Now, as \(h\) approaches 0, the above expression becomes: \[ \lim _{h \rightarrow 0^+} C(100 - h) = \frac{100-1}{2} \] So, \[ \lim _{x \rightarrow 100^-} C(x) = \frac{99}{2} \]

Step 3: Interpret the Limit

The limit obtained, \(\frac{99}{2}\), represents the cost in millions of dollars of removing 100% trichloroethylene in the well. It is important to note that this value is theoretical because as \(x\) gets closer to 100, the cost goes up substantially, and practically, it might not be possible or feasible to reach this limit.

Step 4: Plot the Graph of C(x)

To plot the graph of \(C(x) = \frac{0.5 x}{100 - x}\) within the specified viewing window \([0, 100] \times [0, 10]\), you can use any graphing calculator software or online tools, such as: Desmos, GeoGebra, or a TI-84 calculator. Here's what to do: 1. Open the graphing calculator program or website. 2. Input \(C(x) = \frac{0.5 x}{100 - x}\) 3. Set the viewing window from \(0\) to \(100\) on the x-axis and from \(0\) to \(10\) on the y-axis. 4. You should now see the graph of the cost function within the given window. Observe how the graph increases rapidly as it approaches \(x=100\).

Key Concepts

Rational Functions

Rational functions are mathematical expressions that represent the division of two polynomials. In the problem presented, the function used to estimate the cost of chemical pollution cleanup is given by the formula:
\[C(x) = \frac{0.5x}{100 - x}\].
This representation, where a polynomial in the numerator is divided by a polynomial in the denominator, is a classic example of a rational function. Rational functions often exhibit interesting behaviors as the variables approach certain values that make the denominator zero, which is where the concept of limits becomes very important.

Characteristics of Rational Functions:

• They can have vertical asymptotes, which occur at points where the denominator is zero and the function is undefined.
• They may have horizontal or oblique asymptotes that describe the end behavior as the variable grows infinitely large or small.
• The graph of a rational function often includes hyperbolas or may resemble a combination of separate curves.

Understanding the properties of rational functions can aid in predicting their behavior, such as determining the cost efficiency of removing a pollutant, and in sketching their graph for a more visual interpretation.

Asymptotic Behavior

The concept of asymptotic behavior in mathematics describes how a function behaves as it approaches a certain point or goes to infinity. In the chemical pollution problem described, when evaluating the limit of the cost function as \(x\) approaches 100 from the left, we are essentially exploring the function's asymptotic behavior near the vertical asymptote at \(x=100\).

Understanding Asymptotes:

• A vertical asymptote reflects values that a function approaches but never quite reaches, typically where the function becomes unbounded.
• Horizontal asymptotes represent the value that the function's output gets closer to as the input either increases or decreases without bound.

For the cost function \(C(x)\), an interesting note is that as the percentage of the contaminant being removed approaches 100%, the cost approaches the asymptote, suggesting an infinitely large expense. This can be interpreted as the practical impossibility of removing all contaminants, or that attempting to do so becomes prohibitively expensive. The process of understanding and finding these behaviors is critical in calculus and helps to inform decisions in real-world contexts, like environmental cleanup efforts.

Graphing Functions

Graphing functions is a fundamental aspect of understanding mathematical concepts visually, which helps in grasping both the general form and specific behaviors of functions. When graphing the rational function described in the chemical pollution problem, we can visualize the relationship between the cost and the amount of the pollutant removed.

Plotting a Rational Function:

• Identify the viewing window that frames the important parts of the function - in this case, the window is \([0,100] \times [0,10]\).
• Examine the behavior near any vertical or horizontal asymptotes to understand the limits and end behavior.
• Look for any symmetries or intercepts that might simplify the graphing process.
• Note any regions where the function significantly increases or decreases, as these are key to understanding the function’s overall behavior.

By following these steps and using graphing calculator software like Desmos or GeoGebra, one can accurately render the function's graph. In the given example, this visual aid demonstrates how the cost surges as we approach complete removal of the chemical, which is mathematically expressed through the function's asymptotic behavior. Ultimately, graphing functions serves as a powerful tool for translating complex algebraic concepts into an easily digestible visual format, aiding in comprehensive understanding.