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.

Question

Post a detailed response to the Solving Problems with Functions Discussion prompt. In a student's understanding.

Answer 1

Scenario: Monthly Water Usage Based on Household Size

In this scenario, we are modeling the monthly water usage of a household based on the number of residents living in the house. Water usage typically varies with the size of the household due to factors like increased bathing, laundry, cooking, and outdoor activities.

Definitions of Variables

Let:

• x = number of residents in the household
• W(x) = monthly water usage in gallons

Piecewise Function

We will create a piecewise function to describe how water usage changes based on the number of residents:

1. For households with 1 to 2 residents, the water usage is relatively low and can be modeled as a linear function:

   • If \( x \leq 2 \): \[ W(x) = 50x + 20 \] (This represents a base water usage of 20 gallons plus 50 gallons per resident.)

2. For households with 3 to 4 residents, the water usage increases but starts to stabilize; thus, it’s a linear function with a lower slope:

   • If \( 2 < x \leq 4 \): \[ W(x) = 100 + 40(x - 2) \] (This indicates that after the first two residents, each additional resident uses 40 gallons.)

3. For households with 5 or more residents, the water usage increases more significantly due to higher demands in a larger household:

   • If \( x > 4 \): \[ W(x) = 180 + 60(x - 4) \] (Here, the base usage for 4 residents is 180 gallons, and each additional resident uses 60 gallons.)

Complete Piecewise Function

Putting it all together, the piecewise function looks like this:

\[ W(x) = \begin{cases} 50x + 20 & \text{if } x \leq 2 \ 100 + 40(x - 2) & \text{if } 2 < x \leq 4 \ 180 + 60(x - 4) & \text{if } x > 4 \end{cases} \]

Graph of the Function

To visualize this function, you would plot the three segments on a graph where the x-axis represents the number of residents and the y-axis represents the monthly water usage in gallons.

1. For \( x \leq 2 \):

   • This line starts at (0, 20) and increases to (2, 120).

2. For \( 2 < x \leq 4 \):

   • This line starts at (2, 120) and increases to (4, 180).

3. For \( x > 4 \):

   • This line starts at (4, 180) and increases steeply, depicting more water usage with more residents.

Conclusion

This piecewise function effectively models how water usage in a household tends to increase with the number of residents but does so at varying rates, reflecting real-world habits and needs. By analyzing this function, one can estimate expected water usage for various household sizes, aiding in resource planning and management.