Suppose that the cost (in millions of dollars) to remove x...

Suppose that the cost (in millions of dollars) to remove $x$ percent of a certain pollutant is given by the cost-benefit function $$f(x)=\frac{20 x}{102-x} \text { for } 0 \leq x \leq 100.$$ (a) Find the cost to remove $85 \%$ of the pollutant. (b) Find the cost to remove the final $5 \%$ of the pollutant.

Suppose that the cost (in millions of dollars) to remove $x$ percent of a certain pollutant is given by the cost-benefit function
$$f(x)=\frac{20 x}{102-x} \text { for } 0 \leq x \leq 100.$$
(a) Find the cost to remove $85 \%$ of the pollutant.
(b) Find the cost to remove the final $5 \%$ of the pollutant.

Key Concepts

Function Evaluation

This concept involves substituting a specific input value into a function to determine its output. In the context of a cost function, it means calculating the cost required to achieve a particular level of pollutant removal by directly evaluating the function at that given percent.

Rational Functions

Rational functions, which are ratios of polynomials, often model situations where the relationship between variables is non-linear. They are useful for representing scenarios such as diminishing returns or increasing costs, particularly as inputs approach values that make the denominator small, leading to rapid increases in the function's value.

Incremental vs Cumulative Cost

This concept distinguishes between the overall or cumulative cost to reach a certain level and the additional (incremental) cost needed to move from one level to a higher level. In cost analysis, analyzing the difference between function values at two points helps determine the extra cost associated with achieving a higher degree of pollutant removal.

Environmental Economics Cost-Benefit Analysis

Cost-benefit analysis in environmental economics involves comparing the costs of pollution reduction measures to the benefits derived from cleaner environments. It provides a framework for decision-making by quantifying and analyzing the trade-offs between the expense incurred in pollutant removal and the resulting economic and health benefits.

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