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Think of two real-life quantities that are related to each other and construct a function that models the relationship between the two quantities. The function should be a piecewise function where not all components are linear. Describe the scenario you are modeling, the definitions of the variables involved in the function, and a graph or equation of the function.

Post a detailed response to the Solving Problems with Functions Discussion prompt.

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Scenario: Monthly Cell Phone Plan Costs

In today's world, choosing a cell phone plan can significantly affect monthly expenses. For this exercise, let’s model the relationship between the amount of data used by a customer and their monthly cell phone bill. We define a piecewise function that describes how costs vary based on data usage.

Variables Involved

  1. x: Data used in gigabytes (GB) during a month.
  2. C(x): Total monthly cell phone bill in dollars.

Description of the Function

  • For 0 ≤ x ≤ 2 GB: The customer is on a basic plan with a flat fee of $30. This cost covers up to 2 GB of data without additional charges.

    \[ C(x) = 30 \quad \text{for } 0 \leq x \leq 2 \]

  • For 2 < x ≤ 10 GB: The plan has a tiered pricing model for data usage between 2 GB and 10 GB. After using 2 GB, the customer incurs an additional charge of $10 per GB used over the initial 2 GB.

    \[ C(x) = 30 + 10(x - 2) \quad \text{for } 2 < x \leq 10 \]

  • For x > 10 GB: The customer is then moved to an unlimited data plan that costs a flat rate of $100, regardless of data usage, since they are likely to use a much higher volume of data.

    \[ C(x) = 100 \quad \text{for } x > 10 \]

Complete Piecewise Function

Combining the three segments, we can express the entire cost function \(C(x)\) as:

\[ C(x) = \begin{cases} 30 & \text{for } 0 \leq x \leq 2 \ 30 + 10(x - 2) & \text{for } 2 < x \leq 10 \ 100 & \text{for } x > 10 \end{cases} \]

Graph of the Function

To visualize this function, we can create a graph that shows how the cost \(C(x)\) changes with different amounts of data \(x\).

  1. The first segment (from 0 to 2 GB) is a horizontal line at \(C = 30\).
  2. The second segment (from just over 2 GB to 10 GB) has a slope of 10, starting from the point (2, 30) and ending at (10, 100).
  3. The third segment is a horizontal line starting from the point (10, 100) and continuing indefinitely for any data usage above 10 GB.

Graph Visualization

  • For the 0 to 2 GB range, the graph remains flat at $30.
  • As soon as usage increases beyond 2 GB, the cost begins to rise steeply, displaying the tiered pricing until it levels out again at $100 for unlimited usage.

Conclusion

This piecewise function accurately models the relationship between data usage and monthly billing costs for a cell phone plan, demonstrating how costs rise with increased data use but eventually cap at a predetermined limit for extensive use. This maintains clarity for consumers about their expected charges based on their data consumption, illustrating the nuances of tiered billing structures.