Problem 68
Question
MAKE A DECISION: WATER POLLUTION The cost \(\bar{C}\) (in millions of dollars) of removing \(p\) percent of the industrial and municipal pollutants discharged into a river is \(C=\frac{255 p}{100-p}, \quad 0 \leq p<100\) (a) Find the cost of removing \(15 \%\) of the pollutants. (b) Find the cost of removing \(50 \%\) of the pollutants. (c) Find the cost of removing \(80 \%\) of the pollutants. (d) According to the model, would it be possible to remove \(100 \%\) of the pollutants? Explain.
Step-by-Step Solution
Verified Answer
The costs for removing 15%, 50% and 80% of the pollutants are approximately $45 million, $255 million, and $1020 million, respectively. According to the model, it would not be possible to remove 100% of the pollutants as that would make the cost undefined.
1Step 1: Calculation for 15% removal
Substitute \(p = 15\) into the given cost model: \(C=\frac{255 \cdot 15}{100-15} \) . Solve the expression to get the cost.
2Step 2: Calculation for 50% removal
Substitute \(p = 50\) into the given cost model: \(C=\frac{255 \cdot 50}{100-50} \) . Solve the expression to get the cost.
3Step 3: Calculation for 80% removal
Substitute \(p = 80\) into the given cost model: \(C=\frac{255 \cdot 80}{100-80} \) . Solve the expression to get the cost.
4Step 4: Evaluation of 100% removal
Observe the cost model. If \(p = 100\), the denominator becomes zero, making the cost undefined. Thus, according to this model, it is not possible to remove 100% of the pollutants.
Key Concepts
Percentage Pollutant RemovalMathematical ModellingCost Function Analysis
Percentage Pollutant Removal
Understanding the concept of percentage pollutant removal is crucial when dealing with environmental cleanup projects.
The exercise explores how the percentage of pollutants removed from a river affects the cost involved in the process.
To describe this, a percentage is used to indicate the proportion of pollutants being eliminated from the environment. For instance, removing 15% of river pollutants means that 15 out of every 100 pollutants are extracted.
This concept helps stakeholders comprehend the level of environmental purification achieved. It also enables them to gauge the effort and resources needed to reach specific cleanup targets. Typically, higher removal percentages incur more costs.
This knowledge is vital for setting goals and allowing budgetary considerations when planning pollution control measures.
Here, the formula given for cost helps transform this abstract percentage concept into real-world financial implications.
The exercise explores how the percentage of pollutants removed from a river affects the cost involved in the process.
To describe this, a percentage is used to indicate the proportion of pollutants being eliminated from the environment. For instance, removing 15% of river pollutants means that 15 out of every 100 pollutants are extracted.
This concept helps stakeholders comprehend the level of environmental purification achieved. It also enables them to gauge the effort and resources needed to reach specific cleanup targets. Typically, higher removal percentages incur more costs.
This knowledge is vital for setting goals and allowing budgetary considerations when planning pollution control measures.
Here, the formula given for cost helps transform this abstract percentage concept into real-world financial implications.
Mathematical Modelling
Mathematical modeling offers a powerful tool to describe complex real-world processes through mathematical equations and expressions.
In this exercise, the cost of removing pollutants from an industrial discharge is presented as a function of the percentage of pollutants being removed.
The provided cost model, \(C=\frac{255p}{100-p}\), illustrates a sophisticated example of mathematical modeling. This equation showcases the relationship between the percentage of pollutants removed and the corresponding cost.
By substituting different values of \(p\) (percentage pollutants), the model helps in estimating the cost required for each level of pollutant reduction. Thus, it becomes a practical tool to evaluate financial and environmental trade-offs.
This highlights the importance of mathematical modeling in decision-making processes by simplifying complex data into understandable relationships.
In this exercise, the cost of removing pollutants from an industrial discharge is presented as a function of the percentage of pollutants being removed.
The provided cost model, \(C=\frac{255p}{100-p}\), illustrates a sophisticated example of mathematical modeling. This equation showcases the relationship between the percentage of pollutants removed and the corresponding cost.
By substituting different values of \(p\) (percentage pollutants), the model helps in estimating the cost required for each level of pollutant reduction. Thus, it becomes a practical tool to evaluate financial and environmental trade-offs.
This highlights the importance of mathematical modeling in decision-making processes by simplifying complex data into understandable relationships.
Cost Function Analysis
Analyzing the cost function in pollution removal offers keen insights into budget planning and resource allocation.
The cost function \(C=\frac{255p}{100-p}\) is particularly informative. It reveals an interesting behavior where the cost increases disproportionately as the percentage of pollutants removed approaches 100%. This indicates a key discovery: achieving full removal (100%) is mathematically—and practically—unfeasible since the cost tends towards infinity due to division by zero.
Evaluating Step 4 of the solution makes this clear. At \(p=100\), the denominator \((100-p)\) becomes zero, rendering the function undefined, and thus implying an infinite cost. This analysis provides a reality check for project planners who might strive for idealistic targets.
Through this function, project stakeholders can comprehensively scrutinize and interpret various cost implications against set environmental goals.
Cost function analysis is thus indispensable for developing effective strategies and anticipating financial demands.
The cost function \(C=\frac{255p}{100-p}\) is particularly informative. It reveals an interesting behavior where the cost increases disproportionately as the percentage of pollutants removed approaches 100%. This indicates a key discovery: achieving full removal (100%) is mathematically—and practically—unfeasible since the cost tends towards infinity due to division by zero.
Evaluating Step 4 of the solution makes this clear. At \(p=100\), the denominator \((100-p)\) becomes zero, rendering the function undefined, and thus implying an infinite cost. This analysis provides a reality check for project planners who might strive for idealistic targets.
Through this function, project stakeholders can comprehensively scrutinize and interpret various cost implications against set environmental goals.
Cost function analysis is thus indispensable for developing effective strategies and anticipating financial demands.
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