Pollution The cost C (in millions of dollars) of removing p...

Pollution The cost $C$ (in millions of dollars) of removing $p \%$ of the industrial and municipal pollutants discharged into a river is given by $C=\frac{225 p}{100-p}, \quad 0 \leq p<100$ (a) Use a graphing utility to graph the cost function. (b) Find the costs of removing $10 \%, 40 \%,$ and $75 \%$ of the pollutants. (c) According to this model, would it be possible to remove $100 \%$ of the pollutants? Explain.

Pollution The cost $C$ (in millions of dollars) of removing $p \%$ of the industrial and municipal pollutants discharged into a river is given by
$C=\frac{225 p}{100-p}, \quad 0 \leq p<100$
(a) Use a graphing utility to graph the cost function.
(b) Find the costs of removing $10 \%, 40 \%,$ and $75 \%$ of the pollutants.
(c) According to this model, would it be possible to remove $100 \%$ of the pollutants? Explain.

Key Concepts

Function Evaluation and Limits

Function evaluation involves computing the output for specific input values, which is critical when applying a model to solve real-world problems. Moreover, limits play an essential role in understanding the behavior of a function as the variable approaches a certain value, particularly near points of discontinuity or asymptotes. This concept is crucial for predicting outcomes in scenarios where a function nears, but does not reach, a certain boundary.

Rational Functions

A rational function is defined as the quotient of two polynomials. This type of function often shows characteristics such as vertical and horizontal asymptotes, and it may have restrictions in its domain where the denominator is zero. Recognizing a function as rational helps in predicting its long?term behavior and potential discontinuities.

Domain and Asymptotic Behavior

Understanding the domain of a function involves determining the set of valid input values for which the function is defined. In rational functions, this usually means identifying the values that make the denominator zero, which in turn can lead to vertical asymptotes. An asymptote represents a boundary that the function approaches but never actually touches, providing key insight into the behavior of the function as the variable approaches critical values.

Graphing and Visualization

Graphing utilities allow for visual representation of functions, enabling the analysis of behavior across different regions. For rational functions, graphing can reveal features such as asymptotes, intercepts, and the overall shape of the curve. This visualization aids in understanding how the function behaves especially near discontinuities and over large intervals.

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