Scenario: Water Usage Based on Number of Household Members
In this scenario, we model the relationship between the number of members in a household and their daily water usage in gallons. The function will account for the fact that as the number of people in the household increases, water consumption rises, but the rate of increase may vary due to efficiencies when more members are present.
Definitions of Variables
- \( x \): Number of members in the household (where \( x \geq 1 \)).
- \( y \): Daily water usage in gallons.
Piecewise Function
The water usage may vary based on the number of people in the household:
\[ y(x) = \begin{cases} 50 & \text{if } x = 1 \ 50 + 30(x - 1) & \text{if } 2 \leq x \leq 4 \ 50 + 30(3) + 20(x - 4) & \text{if } x > 4 \end{cases} \]
Explanation of Each Case
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For \( x = 1 \): A single person uses a baseline of 50 gallons per day.
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For \( 2 \leq x \leq 4 \): Each additional member (from the 2nd to the 4th) adds approximately 30 gallons more per day due to shared activities such as showering, cooking, etc. Thus, for 2 members, the water usage is 80 gallons, for 3 members it is 110 gallons, and for 4 members, it is 140 gallons.
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For \( x > 4 \): Beyond 4 members, the daily water usage increases by 20 gallons for each additional member. This reflects a more efficient use of water resources and space—shared showers, laundry, and dishwashing—resulting in less than linear growth. For example, for 5 members, it would be \( 140 + 20(1) = 160 \) gallons; for 6 members, it would be \( 140 + 20(2) = 180 \) gallons, and so on.
Graph of the Function
The graph of this piecewise function can be drawn in the coordinate system, showing the flat baseline for one person, a steep incline for the next three members, and a gradually ascending line with a smaller slope for households larger than four members.
- For \( x = 1 \), the point is at (1, 50).
- For \( x = 2 \), the point at (2, 80).
- For \( x = 3 \), the point at (3, 110).
- For \( x = 4 \), the point at (4, 140).
- For \( x = 5 \), (5, 160).
- For \( x = 6 \), (6, 180).
Conclusion
This piecewise function effectively models how water usage changes with the size of a household, demonstrating both an initial steep growth and then a slower growth pattern as the household size increases.