 1. Mary's water bill changes as the amount of water used. If Mary uses less than 100 gallons of water the cost is 10 cents a gallon ($0.10). If Mary uses 100 gallons to 500 gallons the cost is 15 cents ($0.15 ) a gallon plus a $20 fee to encourage conservation If Mary uses over 500 gallons the cost is 20 cents ($ 0.20) a gallon plus a $50 fee to encourage conservation

This problem can be modeled using a piecewise function based on the amount of water used. Here's the breakdown:

### Given Information:
- **Less than 100 gallons**: Cost is $0.10 per gallon.
- **Between 100 and 500 gallons**: Cost is $0.15 per gallon plus a $20 fee.
- **Over 500 gallons**: Cost is $0.20 per gallon plus a $50 fee.

### Let $ g $ represent the number of gallons used, and $ C(g) $ represent the cost. The piecewise function for the total cost can be written as:

\[
C(g) = 
\begin{cases} 
0.10g & \text{if } 0 \leq g < 100 \\
0.15g + 20 & \text{if } 100 \leq g \leq 500 \\
0.20g + 50 & \text{if } g > 500
\end{cases}
\]

### Explanation of the Function:
1. For $ g < 100 $: The cost is simple, just $ 0.10g $.
2. For $ 100 \leq g \leq 500 $: In this range, there's an added conservation fee of $20 plus a higher per-gallon charge of $0.15.
3. For $ g > 500 $: The cost jumps to $0.20 per gallon, plus a $50 fee.

Would you like me to help with specific examples, like calculating the cost for certain amounts of water usage?

Here are 5 related questions that expand on this:
1. What would the total cost be if Mary uses 150 gallons of water?
2. How much does Mary pay if she uses exactly 500 gallons of water?
3. What is the price for using 600 gallons of water?
4. At what point does the cost exceed $100?
5. How much more would Mary pay for using 600 gallons compared to 500 gallons?

**Tip**: Piecewise functions are a great way to model situations where different rules apply to different ranges of input values.

Mary's water bill changes as the amount of water used. If Mary uses less than 100 gallons of water the cost is 10 cents a gallon ($0.10). If Mary uses 100 gallons to 500 gallons the cost is 15 cents ($0.15 ) a gallon plus a $20 fee to encourage conservation. If Mary uses over 500 gallons the cost is 20 cents ($0.20) a gallon plus a $50 fee to encourage conservation.

Understand how to calculate water bills using piecewise functions based on gallon usage. This problem demonstrates the use of linear equations and cost modeling, with breakdowns for water usage under 100 gallons, between 100 and 500 gallons, and above 500 gallons. Suitable for Grades 9-11.

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