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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: Shipping Cost Based on Weight of Packages

In this scenario, we are modeling the cost of shipping packages based on their weight. Different shipping carriers have different pricing tiers based on weight, which can lead to non-linear relationships in the cost function.

Definitions of Variables:

  • \( x \): The weight of the package (in pounds).
  • \( C(x) \): The cost to ship the package (in dollars).

Piecewise Function Model:

The shipping cost will be defined as follows:

  1. For packages weighing up to 5 pounds:

    • Up to 5 pounds, the shipping cost is charged at a flat rate of $5. \[ C(x) = 5 \quad \text{for } 0 < x \leq 5 \]

  2. For packages weighing more than 5 pounds but up to 20 pounds:

    • For weights between 5 and 20 pounds, the cost is calculated using a base charge of $5 plus $1 for every additional pound over 5. \[ C(x) = 5 + (x - 5) \quad \text{for } 5 < x \leq 20 \]

  3. For packages weighing more than 20 pounds:

    • For weights greater than 20 pounds, the cost is $20 plus $0.50 for every pound over 20. \[ C(x) = 20 + 0.5(x - 20) \quad \text{for } x > 20 \]

Piecewise Function Summary:

Putting this all together, the cost function \( C(x) \) can be expressed as: \[ C(x) = \begin{cases} 5 & \text{if } 0 < x \leq 5 \ 5 + (x - 5) & \text{if } 5 < x \leq 20 \ 20 + 0.5(x - 20) & \text{if } x > 20 \end{cases} \]

Graph of the Function:

To graph this function:

  1. From \( (0, 5) \) to \( (5, 5) \): A horizontal line at \( y = 5 \).
  2. From \( (5, 5) \) to \( (20, 20) \): A straight line that rises with a slope of 1 (from the equation \( C(x) = 5 + (x - 5) \)).
  3. From \( (20, 20) \): Starts at \( (20, 20) \) and has a slope of 0.5, rising to \( (25, 22.5) \) and continues to rise more gradually.

Visualization:

The graph would look like this:

  • There will be a horizontal line segment from \( (0, 0) \) to \( (5, 5) \).
  • A diagonal line segment rising steeply until it levels off after 20 pounds, transitioning to a gentler incline for weights greater than 20 pounds.

This piecewise function effectively captures the variation in shipping costs depending on the weight of the package, illustrating how shipping costs can change non-linearly based on weight.