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The parking fee problem can be modeled as a piecewise function \( P(h) \), where \( h \) is the number of hours parked. Let's break it down:

- For the first 2 hours, the fee is fixed at Php 35.00.
- For any additional hour beyond the first 2 hours but less than or equal to 15 hours, the cost is Php 10.00 per hour.
- If the total parking duration exceeds 15 hours, the fee becomes a flat rate of Php 180.00.

Now, we can define the function \( P(h) \) as follows:

\[
P(h) =
\begin{cases}
35, & \text{if } 0 < h \leq 2 \\
35 + 10(h - 2), & \text{if } 2 < h \leq 15 \\
180, & \text{if } h > 15
\end{cases}
\]

This function covers all three cases based on the number of hours \( h \) parked.

### Explanation:
- For \( h \leq 2 \): The parking fee is fixed at Php 35.
- For \( 2 < h \leq 15 \): The parking fee starts at Php 35 for the first two hours, plus Php 10 for each additional hour beyond the first two. This is represented by \( 35 + 10(h - 2) \).
- For \( h > 15 \): The fee is a flat Php 180, regardless of how many hours you park beyond 15 hours.

Would you like more details or have any questions?

### Related Questions:
1. How would the function change if the flat rate for parking over 15 hours were Php 200 instead of Php 180?
2. How can you express the function for different rates, like Php 8 for each additional hour after the first two?
3. How do you handle cases where the parking fee changes by fractions of hours instead of whole hours?
4. How would the parking fee function look if the first hour were free?
5. How could you adjust the function if parking for more than 10 hours triggered a different flat rate?

**Tip:** Piecewise functions are ideal for situations where different rules apply over different intervals, such as time-based rates or tiered pricing systems.

A parking fee at SM San Mateo costs Php 35.00 for the first two hours and an extra Php 10.00 for each hour of extension. If you park for more than fifteen hours, you instead pay a flat rate of Php 180.00. Represent your parking fee using the function P(h) where h is the number of hours you parked in the mall.

Learn how to calculate parking fees at SM San Mateo using a piecewise function. The problem breaks down the charges for parking up to 15 hours and applies a flat rate for parking beyond 15 hours. Ideal for high school students learning about piecewise and linear functions.

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