To represent the fare structure for a jeepney ride using a piecewise function, we can break down the fare calculation based on the distance traveled.

Let's assume:
- \( d \) is the distance traveled in kilometers.
- The fare structure is as follows:
  1. **Base fare**: ₱12.00 for the first 4 kilometers.
  2. **Additional fare**: ₱2.00 per kilometer after the first 4 kilometers.

The fare \( f(d) \) as a function of the distance \( d \) can be written as a piecewise function:

\[
f(d) = 
\begin{cases} 
12 & \text{if } 0 \leq d \leq 4 \\
12 + 2(d - 4) & \text{if } d > 4
\end{cases}
\]

This function can be explained as follows:
- If the distance \( d \) is 4 kilometers or less, the fare is a flat ₱12.00.
- If the distance \( d \) is greater than 4 kilometers, the fare starts at ₱12.00 and increases by ₱2.00 for every additional kilometer beyond 4 kilometers.

Now, we can determine the maximum distance \( d \) that can be traveled with a budget of ₱18.00 by solving the piecewise function for \( f(d) = 18 \).

### Case 1: \( 0 \leq d \leq 4 \)
Here, the fare is always ₱12.00, which is less than ₱18.00. Therefore, this case doesn't use up the entire budget.

### Case 2: \( d > 4 \)
\[
f(d) = 12 + 2(d - 4) = 18
\]

Solve for \( d \):
\[
12 + 2(d - 4) = 18
\]
\[
2(d - 4) = 18 - 12
\]
\[
2(d - 4) = 6
\]
\[
d - 4 = \frac{6}{2} = 3
\]
\[
d = 7
\]

So, with a budget of ₱18.00, the maximum distance you can travel is **7 kilometers**.

Would you like more details or have any other questions?

Here are five related questions you might find interesting:
1. How would the piecewise function change if the base fare increased?
2. What if the distance included additional flat-rate charges beyond the first 10 kilometers?
3. How would a budget of ₱20.00 affect the maximum travel distance?
4. Can the piecewise function be extended to account for discounts (e.g., for students or seniors)?
5. How would the fare structure change if the additional fare per kilometer decreased?

**Tip**: Understanding how to model real-world problems using piecewise functions can help solve various budgeting and cost-related issues efficiently.

Explore how to use piecewise functions to calculate jeepney fare based on distance traveled. Learn step-by-step how to model and solve for maximum travel distance with a budget of ₱18.00 using the given fare structure. Suitable for high school students studying piecewise functions and practical applications of linear equations.

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