The cost C (in dollars) of supplying recycling bins to p %...

The cost $C$ (in dollars) of supplying recycling bins to $p \%$ of the population of a rural township is given by $$C=\frac{25,000 p}{100-p}, \quad 0 \leq p<100$$ (a) Use a graphing utility to graph the cost function. (b) Find the costs of supplying bins to $15 \%$ $50 \%,$ and $90 \%$ of the population. (c) According to the model, would it be possible to supply bins to $100 \%$ of the population? Explain.

The cost $C$ (in dollars) of supplying recycling bins to $p \%$ of the population of a rural township is given by $$C=\frac{25,000 p}{100-p}, \quad 0 \leq p<100$$
(a) Use a graphing utility to graph the cost function.
(b) Find the costs of supplying bins to $15 \%$ $50 \%,$ and $90 \%$ of the population.
(c) According to the model, would it be possible to supply bins to $100 \%$ of the population? Explain.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in rational functions at values that cause the denominator to zero, resulting in the function's value growing without bound. They indicate critical thresholds beyond which the function's behavior becomes extreme. In practical models, such as this cost model, vertical asymptotes signal points where the model breaks down or becomes impractical, emphasizing limitations such as not being able to supply bins to a full 100% of the population.

Rational Functions

A rational function is defined as the quotient of two polynomials. In this context, the cost function is expressed as a ratio, where both the numerator and denominator depend on the variable representing a percentage. Understanding rational functions helps in analyzing how the cost behaves as the parameter changes, especially when the denominator approaches zero or when the degree of the numerator and denominator affect the overall behavior of the function.

Domain Restrictions

The domain of a function consists of all possible input values for which the function is well-defined. For rational functions, values that make the denominator zero must be excluded, as they lead to undefined expressions. In modeling scenarios like this cost function, recognizing the domain restrictions is crucial to avoid unrealistic or non-physical outputs, such as supplying recycling bins to an impossible percentage of the population.

Function Evaluation

Function evaluation involves substituting specific values into the function to compute the corresponding outputs. This process is essential in applied problems where particular input values—like specific percentages of a population—are analyzed to determine associated outcomes, such as the cost. It provides concrete results that help validate and illustrate the behavior predicted by the mathematical model.

Graphical Analysis

Graphical analysis uses visual representations to understand the behavior of functions over their entire domain. By graphing the cost function, one can easily observe trends, intercepts, and asymptotic behavior. This approach aids in interpreting the model's predictions and facilitates a better understanding of how changes in the percentage of the population affect the overall cost, as well as highlighting the limitations inherent in the model.

Please give Ace some feedback