A large city has initiated a new recycling effort, and wants...

A large city has initiated a new recycling effort, and wants to distribute recycling bins for use in separating various recyclable materials. City planners anticipate the cost of the program can be modeled by the function $C(p)=\frac{220 p}{100-p}$ where $C(p)$ represents the cost (in $\$ 10,000$ ) to distribute the bins to $p$ percent of the population. (a) Find the cost to distribute bins to $25 \%, 50 \%$ and $75 \%$ of the population, then comment on the results; (b) graph the function using an appropriate scale; and (c) use the direction/ approach notation to state what happens if the city attempts to give recycling bins to $100 \%$ of the population.

A large city has initiated a new recycling effort, and wants to distribute recycling bins for use in separating various recyclable materials. City planners anticipate the cost of the program can be modeled by the function $C(p)=\frac{220 p}{100-p}$ where $C(p)$ represents the cost (in $\$ 10,000$ ) to distribute the bins to $p$ percent of the population.
(a) Find the cost to distribute bins to $25 \%, 50 \%$ and $75 \%$ of the population, then comment on the results; (b) graph the function using an appropriate scale; and (c) use the direction/ approach notation to state what happens if the city attempts to give recycling bins to $100 \%$ of the population.

Key Concepts

Graphing Functions

Graphing functions is a key method in visualizing and analyzing the behavior of mathematical relations. Graphs illustrate important features such as trends, intercepts, and discontinuities, allowing planners to see how a cost function behaves over various levels of service coverage.

Limits and Asymptotic Behavior

The concept of limits is used to describe the behavior of functions as the input approaches a specific point. In many rational functions, as the input nears a value that makes the denominator zero (often creating a vertical asymptote), the function's output may approach infinity. This idea is crucial when discussing the feasibility or extreme cost outcomes in models, such as attempting to serve the entire population in the recycling initiative.

Rational Functions

A rational function is defined as the quotient of two polynomials. In these types of functions, the behavior of the function is heavily influenced by the denominator, especially where it equals zero. This concept is central for analyzing cost models, like in the recycling bins distribution, where the function governs how the cost increases as the percentage of distribution approaches a critical threshold.

Function Evaluation

Function evaluation involves substituting specific input values into a function and simplifying to determine the output. This is a fundamental concept in mathematics that helps in understanding how changes in the input affect the outcome, such as computing the cost for different percentages of service coverage.

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