Problem 61
Question
A city's main well was recently found to be contaminated with trichloroethylene (a cancer-causing chemical) as a result of an abandoned chemical dump that leached chemicals into the water. A proposal submitted to the city council indicated that the cost, measured in millions of dollars, of removing \(x \%\) of the toxic pollutants is given by $$ C(x)=\frac{0.5 x}{100-x} $$ a. Find the vertical asymptote of \(C(x)\). b. Is it possible to remove \(100 \%\) of the toxic pollutant from the water?
Step-by-Step Solution
Verified Answer
a. The vertical asymptote of the cost function \(C(x)=\frac{0.5 x}{100-x}\) is at \(x = 100\).
b. No, it is not possible to remove 100% of the toxic pollutant from the water, as the cost becomes infinite at the vertical asymptote \(x = 100\).
1Step 1: Identify the function and the domain
The cost function is given as \(C(x)=\frac{0.5 x}{100-x}\). This is a rational function, and rational functions have vertical asymptotes where the denominator is equal to zero.
2Step 2: Find the value of x where the denominator is equal to zero
Set the denominator equal to zero and solve for x:
\(100 - x = 0\)
Add x to both sides:
\(100 = x\)
Therefore, the vertical asymptote is at \(x = 100\).
#b. Is it possible to remove 100% of the toxic pollutant from the water?#
3Step 3: Analyze the function at the vertical asymptote
If the cost function is undefined for a certain value of x, it means that the cost becomes infinite. In our case, the vertical asymptote is at \(x = 100\), which means that the cost of removing 100% of the toxic pollutants would be infinite.
4Step 4: Conclude if it is possible to remove 100% of the toxic pollutant
Since the cost of removing 100% of the toxic pollutants is infinite, it is not possible to remove 100% of the toxic pollutant from the water.
Key Concepts
Vertical AsymptoteEnvironmental MathematicsPollution Removal Costs
Vertical Asymptote
In mathematics, a vertical asymptote is a line that a graph approaches but never touches or crosses. For rational functions, like the cost function in our exercise, vertical asymptotes occur at the values of \(x\) that make the denominator zero.
The exercise provides us with the cost function: \[ C(x)=\frac{0.5x}{100-x} \] Here, the denominator \(100-x\) equals zero when \(x=100\). Therefore, there is a vertical asymptote at \(x=100\). This mathematical property indicates that as \(x\) approaches 100, the cost \(C(x)\) will escalate towards infinity, reflecting something significant about the function's behavior. Vertical asymptotes thus help identify values where rational functions behave "abnormally," often becoming infinite or undefined.
The exercise provides us with the cost function: \[ C(x)=\frac{0.5x}{100-x} \] Here, the denominator \(100-x\) equals zero when \(x=100\). Therefore, there is a vertical asymptote at \(x=100\). This mathematical property indicates that as \(x\) approaches 100, the cost \(C(x)\) will escalate towards infinity, reflecting something significant about the function's behavior. Vertical asymptotes thus help identify values where rational functions behave "abnormally," often becoming infinite or undefined.
Environmental Mathematics
Environmental mathematics involves applying mathematical concepts to solve environmental issues. It plays a vital role in understanding and resolving problems related to pollution, conservation, and sustainability.
In our example of pollution removal costs, understanding the cost function is an application of environmental mathematics. This function tells us the financial implications of removing various percentages of toxic pollutants from contaminated water. By calculating potential costs, city planners and environmentalists can make informed decisions. They can balance both ecological and financial aspects to devise effective pollution control strategies that ensure public health and environmental safety.
In our example of pollution removal costs, understanding the cost function is an application of environmental mathematics. This function tells us the financial implications of removing various percentages of toxic pollutants from contaminated water. By calculating potential costs, city planners and environmentalists can make informed decisions. They can balance both ecological and financial aspects to devise effective pollution control strategies that ensure public health and environmental safety.
- Helps in resource allocation
- Predicts future costs
- Aids in policy-making
Pollution Removal Costs
Pollution removal costs are essential for planning and implementing environmental cleanup strategies. In our problem scenario, the cost function \(C(x) = \frac{0.5 x}{100 - x}\) accounts for the financial requirements to clean a certain percentage \(x\) of pollutants.
As \(x\) increases, so does the cost, particularly as it nears 100%. The function shows that as more pollutants are removed, the cost rises sharply due to the vertical asymptote at \(x = 100\). This suggests that while it's feasible to remove most pollutants, removing every last bit (100%) leads to impractical or infinite costs. Such a concept reveals the challenge in environmental cleanups: achieving complete pollution eradication is often not economically viable. Hence, understanding pollution removal costs helps stakeholders recognize the limits of current technology and economics in addressing environmental problems.
As \(x\) increases, so does the cost, particularly as it nears 100%. The function shows that as more pollutants are removed, the cost rises sharply due to the vertical asymptote at \(x = 100\). This suggests that while it's feasible to remove most pollutants, removing every last bit (100%) leads to impractical or infinite costs. Such a concept reveals the challenge in environmental cleanups: achieving complete pollution eradication is often not economically viable. Hence, understanding pollution removal costs helps stakeholders recognize the limits of current technology and economics in addressing environmental problems.
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