Question

Environment The predicted cost \(C\) (in hundreds of thousands of dollars) for a company to remove \(p \%\) of a chemical from its waste water is shown in the table. \begin{tabular}{|c|c|c|c|c|l|} \hline\(p\) & 0 & 10 & 20 & 30 & 40 \\ \hline\(C\) & 0 & \(0.7\) & \(1.0\) & \(1.3\) & \(1.7\) \\ \hline \end{tabular} \begin{tabular}{|l|l|l|l|l|l|} \hline\(p\) & 50 & 60 & 70 & 80 & 90 \\ \hline\(C\) & \(2.0\) & \(2.7\) & \(3.6\) & \(5.5\) & \(11.2\) \\ \hline \end{tabular} A model for the data is given by \(C=\frac{124 p}{(10+p)(100-p)}, \quad 0 \leq p<100\) Use the model to find the average cost for removing between \(75 \%\) and \(80 \%\) of the chemical.

Short answer

Use the steps above to find the answer. You might need to use the software or a calculator to evaluate the definite integral in Step 3, as the integral could be complicated to solve step by step. The final answer will be the average cost in hundreds of thousands dollars for the company to remove between 75% and 80% of the chemical from its waste water.

Step-by-step solution

Understand the Model

The provided model is a function that associates a percentage \(p\) to a cost \(C\). The variable \(p\) represents the percentage of a chemical that the company aims to remove from its waste water.

Setting Up The Integral

Firstly, convert the percentage into decimals. To find the average cost of removing between 75% to 80%, we need to evaluate the definite integral of the function from 0.75 to 0.80. We then divide the result by the length of the interval (0.80 - 0.75) to find the average. This can be expressed as \[average = \frac{1}{0.80 - 0.75}\int_{0.75}^{0.80} \frac{124p}{(10+p)(100-p)} dp\]

Evaluate The Definite Integral

Use a suitable method to find the value of definite integral. For some integrals, it might be useful to use software or calculator to find the exact result.

Determine Average Cost

Multiply the result obtained in the previous step by the reciprocal of the length of the interval (which is 1/(0.80 - 0.75))

Key concepts

Definite Integral Application

Calculus, and specifically the concept of the definite integral, is a powerful tool in various scientific disciplines, one being economics. In the context of environmental economics, the definite integral finds practical application in calculating the average cost, total accumulated quantities, and other significant values related to cost and resource management.

For instance, if a company needs to determine the average cost of a process over a certain interval, like removing a percentage of chemicals from wastewater, the definite integral proves essential. By setting up an integral over the interval of interest and dividing by the interval's length, the average value within that segment can be efficiently calculated. This application not only simplifies complex problems but also provides precise and valuable insights into economic decisions.

Integral Calculus in Environmental Science

The role of integral calculus in environmental science is versatile and far-reaching. By integrating mathematical concepts with environmental data, it enables researchers and policymakers to model and predict various scenarios and manage resources sustainably.

For example, integral calculus helps to quantify the environmental impact of certain substances, like chemicals in wastewater. By exercising the integral calculus on empirical data, or predictive models, scientists can estimate the amount of pollutants removed over an interval, the total impact on local ecosystems, or track changes in environmental quality.

Environmental Impact Modelling

By creating models based on integral calculus, scientists can simulate different scenarios, such as the progression of pollutants over time, which aids in forming strategies for environmental protection and resource management.

Mathematical Modeling

Mathematical modeling is akin to painting a picture, except the paints are numbers and equations, and the canvas is our understanding of the world around us.

In essence, mathematical modeling employs mathematical language to describe relationships and changes within systems, whether they be physical, biological, chemical, or economic. In environmental science, a model might showcase how different variables - such as the percentage of a pollutant removed - affect the cost for a company.

Such models, like the one provided in the exercise, offer a simplified yet accurate representation of reality, which can serve as a predictive tool for decision-making. While these models are essential for anticipating outcomes and planning, it's important to remember they rely on assumptions and simplifications. Despite these limitations, mathematical models are a cornerstone in a multitude of fields, reflecting the need for systematic and quantitative reasoning in tackling complex real-world problems.