A parking fee at SM Consolacion costs P25.00 for the first two hours and an extra P5.00 for each hour of extension. If you park for more than twelve hours, you instead pay a flat rate of P100.00. Represent your parking fee using the function p(t) where t is the number of hours you parked in the mall.

To represent the parking fee as a function \( p(t) \), where \( t \) is the number of hours parked, you can break it down into three cases:

1. **Case 1:** For parking between 0 and 2 hours (inclusive):
   - The fee is a flat rate of \( P25.00 \).

2. **Case 2:** For parking more than 2 hours but not exceeding 12 hours:
   - The fee is \( P25.00 \) for the first 2 hours plus \( P5.00 \) for each additional hour.
   - This can be calculated as \( P25 + 5(t - 2) \) where \( t > 2 \).

3. **Case 3:** For parking more than 12 hours:
   - The fee is a flat rate of \( P100.00 \).

The piecewise function can be represented as:

\[
p(t) = 
\begin{cases} 
25 & \text{if } 0 \leq t \leq 2 \\
25 + 5(t - 2) & \text{if } 2 < t \leq 12 \\
100 & \text{if } t > 12 
\end{cases}
\]

### Explanation:
- For \( 0 \leq t \leq 2 \): The function gives a flat rate of \( P25.00 \).
- For \( 2 < t \leq 12 \): The function starts with \( P25.00 \) and adds \( P5.00 \) for each hour beyond the initial 2 hours.
- For \( t > 12 \): The function outputs \( P100.00 \), which is the flat rate for parking more than 12 hours.

Would you like more details or have any questions?

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### Related Questions:
1. How would the function change if the flat rate after 12 hours were \( P120.00 \) instead of \( P100.00 \)?
2. What is the parking fee for someone who parked for 4 hours?
3. What is the domain and range of the function \( p(t) \)?
4. How can you graph the function \( p(t) \)?
5. What if the extra charge after 2 hours were \( P10.00 \) per hour instead of \( P5.00 \)? How would the function change?
6. How much would it cost to park for 8 hours?
7. How do you calculate the average rate of change of the parking fee between 2 and 6 hours?
8. If the mall decided to offer a discount of \( P10.00 \) for parking less than 6 hours, how would the function be modified?

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### Tip:
When working with piecewise functions, always clearly define the conditions for each piece and ensure the function is continuous if necessary.

Learn how to represent parking fees using piecewise functions. This problem explores different rates for parking at SM Consolacion, including flat rates and hourly extensions. Suitable for Grades 7-9 students studying piecewise functions and linear equations.

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