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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...............

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Scenario

Let's model the cost of electricity consumption based on the number of kilowatt-hours (kWh) used by a household. Electricity costs often include a flat base fee plus a variable cost based on usage. For small amounts of usage, the cost is calculated at a lower rate, while it increases at a higher rate for higher usage. Additionally, to encourage energy conservation, a higher price tier kicks in after a certain threshold of consumption.

Variables Defined

  • \( x \): The number of kilowatt-hours (kWh) consumed by the household.
  • \( C(x) \): The total cost of electricity in dollars based on the consumption \( x \).

Piecewise Function Definition

The cost function \( C(x) \) can be defined as follows:

\[ C(x) = \begin{cases} 10 + 0.05x & \text{if } 0 \leq x \leq 200 \ 20 + 0.10(x - 200) & \text{if } 200 < x \leq 600 \ 80 + 0.20(x - 600) & \text{if } x > 600 \end{cases} \]

Description of the Function Components

  1. For Consumption up to 200 kWh:

    • If a household consumes from 0 to 200 kWh, they pay a flat fee of $10, plus $0.05 for every kWh consumed.

    Equation: \( C(x) = 10 + 0.05x \)

  2. For Consumption from 201 to 600 kWh:

    • If the consumption exceeds 200 kWh but is at most 600 kWh, they pay the base fee of $20 for the first 200 kWh, plus $0.10 for each additional kWh over 200.

    Equation: \( C(x) = 20 + 0.10(x - 200) \)

  3. For Consumption Above 600 kWh:

    • For consumption greater than 600 kWh, the cost starts with a base of $80 (which covers the previous tiers), and then charges $0.20 for each additional kWh over 600.

    Equation: \( C(x) = 80 + 0.20(x - 600) \)

Graph of the Function

The graph of this piecewise function would show the total cost \( C(x) \) on the y-axis and the kWh \( x \) on the x-axis. Here’s how the components would look graphically:

  • From \( x = 0 \) to \( x = 200 \), the line would rise gently.
  • Once it hits \( x = 200 \), the slope of the line would increase, reflecting the higher cost per kWh.
  • At \( x = 600 \), the slope increases again to reflect the even higher rate of $0.20 per kWh.

Summary

In summary, this piecewise function models how household electricity costs escalate based on usage, promoting energy conservation by introducing higher rates for higher consumption. The specific rates and thresholds can vary in real-world scenarios, but this example demonstrates how a piecewise function can effectively model non-linear cost structures in real-life situations.