Problem 82 - College Algebra Essentials

Question

write a piecewise function that models each cellphone billing plan. Then graph the function. \(\$ 60\) per month buys 450 minutes. Additional time costs \(\$ 0.35\) per minute.

Step-by-Step Solution

Answer

The piecewise function that models the given cellphone billing plan is \( C(m) = \begin{cases} 60 & \text{for } m \leq 450 \ 60 + 0.35(m - 450) & \text{for } m > 450 \end{cases} \). The graph begins at \( C(m) = 60 \) and remains at this level until \( m=450 \), then it increases with a slope of \( $0.35 \) per minute.

Step 1: Define the Piecewise Function

The cost of the cell phone plan can be represented as a piecewise function. For the first 450 minutes, the cost is \( $60 \) per month, hence the first equation is \( C(m) = 60 \) for \( m<=450 \). After 450 minutes, each additional minute costs \( $0.35 \), hence the second equation for \( m>450 \) becomes \( C(m) = 60 + 0.35*(m-450) \). Putting these together, the piecewise function becomes \( C(m) = \begin{cases} 60 & \text{for } m \leq 450 \ 60 + 0.35(m - 450) & \text{for } m > 450 \end{cases} \).

Step 2: Graph the Piecewise Function

The function is broken into two sections: 1) \( m<=450 \), which is a horizontal line on the graph at \( C(m) = 60 \), and 2) \( m>450 \), which has a slope of \( $0.35 \) and begins at \( m=450 \), \( C(m) = 60 \). In the section for \( m>450 \), for every minute increase the cost goes up by \( $0.35 \).

Key Concepts

Cellphone Billing Plans

Cellphone billing plans often include a base package and additional costs for extra usage. In the example provided, we see a common structure where a standard number of minutes is offered for a fixed monthly fee. Here, for $60 per month, you are allowed to use up to 450 minutes. This base cost covers the typical usage of a consumer. However, if you happen to use more than the allotted 450 minutes, additional charges apply. In this scenario, each extra minute costs $0.35. It is important to carefully check these details when choosing a plan to avoid unexpected charges. Understanding how these charges accumulate helps you to predict your monthly expenses effectively. When additional charges exist, it's crucial to monitor your usage closely or consider switching to a plan that better fits your needs to avoid overage fees. This principle of structured billing using a base plus incremental charge is common beyond cellphone plans, and understanding it can be financially beneficial.

Mathematical Modeling

Mathematical modeling allows us to represent real-world scenarios with mathematical expressions. In this case, the cellphone billing plan is translated into a piecewise function. This helps to clearly define cost scenarios based on usage.The piecewise function here consists of two "pieces" or expressions. For usage less than or equal to 450 minutes, the cost is a flat \(60. This is represented as \( C(m) = 60 \). For minutes exceeding 450, the cost increases by \)0.35 per additional minute, captured by the function \( C(m) = 60 + 0.35(m - 450) \).

• Piecewise functions provide an efficient way to model real-world billing scenarios.
• They allow us to create tailored expressions based on specific conditions or situations.

By using a piecewise function, we can articulate the transition between flat rates and additional costs, which also simplifies the interpretation of how cost varies with changes in usage.

Graphing Functions

Graphing functions is a useful technique for visualizing how variables interact. In the context of the cellphone billing plan, graphing allows us to see how the cost changes with the number of minutes used.In this exercise, the piecewise function is graphed in two segments:

• The first segment represents the flat cost of \(60 for up to 450 minutes, a horizontal line on the graph.
• Once usage exceeds 450 minutes, the cost increases linearly by \)0.35 per minute over 450, resulting in a line with a positive slope.

To create this graph, plot each segment distinctly:

• For \( m \leq 450 \), draw a straight horizontal line at $60.
• For \( m > 450 \), use the slope-intercept form \( C(m) = 0.35m - 97.5 \), which starts where the flat rate ends.

Graphing these segments together provides insight into the cost structure. It visually demonstrates at which point additional charges start applying, helping consumers understand potential cost implications.