A parking fee at SM lucena cost 25 for the first two hours and an extra 5 for each hour of extension, if you park more than 12 hours you instead pay a flat rate of 100 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 \( p(t) \) at SM Lucena based on the number of hours \( t \) you parked, we need to consider the different cases described:

1. For the first two hours, the fee is a fixed rate of 25 pesos.
2. For every additional hour beyond the first two, there is an additional charge of 5 pesos per hour.
3. If the total parking time exceeds 12 hours, a flat rate of 100 pesos is applied.

Let's define the function \( p(t) \):

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

Breaking this down:

1. For \( 0 < t \leq 2 \), the cost is 25 pesos.
2. For \( 2 < t \leq 12 \), the cost is calculated as:
   \[
   p(t) = 25 + 5(t - 2)
   \]
   Here, \( 5(t - 2) \) represents the additional cost for the extra hours beyond the first two hours.
3. For \( t > 12 \), the cost is a flat rate of 100 pesos.

Thus, the function \( p(t) \) can be written as:

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

### Examples

- If you park for 1 hour, the cost is \( p(1) = 25 \) pesos.
- If you park for 5 hours, the cost is \( p(5) = 25 + 5(5 - 2) = 25 + 15 = 40 \) pesos.
- If you park for 13 hours, the cost is \( p(13) = 100 \) pesos.

Would you like more details or have any questions?

### Related Questions
1. How would the function change if the flat rate for parking more than 12 hours were 120 pesos instead of 100?
2. How would the cost function be affected if the first two hours were free and charges only applied afterward?
3. What would the function look like if the rate after the first two hours was 10 pesos per hour instead of 5 pesos?
4. How would you express the function if the parking fee increased by 1 peso every hour after the first two hours?
5. How would you calculate the parking fee if you parked for exactly 12 hours?
6. How can you use this function to determine the parking fee for any given hour?
7. If the mall offered a discount for parking between 12 PM and 2 PM, how would that be incorporated into the function?
8. How would the function change if the mall capped the maximum parking fee at 80 pesos, regardless of hours parked?

### Tip
When analyzing piecewise functions, always carefully define the conditions for each piece to ensure the function correctly models the given scenario.

Understand how to calculate parking fees at SM Lucena based on hours parked using a piecewise function. Learn about the fixed rate for the first two hours, additional charges per hour thereafter, and the flat rate applied after 12 hours. Example calculations provided.

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