Logistics and Transportation

Leona will go to a mall which is & kilometers from her house and the bus fare costs 12 pesos for the first 4 kilometers and each additional integer kilometer adds up lone Desios to the fare. Use a piecewisend each ad depresent the bus fare in terms of the distance (d) in kilometers.

To represent the bus fare in terms of the distance \( d \) in kilometers, we can define a piecewise function. Here's the breakdown of the fare calculation:

1. For the first 4 kilometers, the bus fare is a fixed 12 pesos.
2. For each additional kilometer beyond the initial 4 kilometers, 1 peso is added to the fare.

Given this information, the bus fare \( F(d) \) in terms of distance \( d \) can be represented as a piecewise function:

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

### Explanation:
- **Case 1:** When \( d \leq 4 \), the distance is within the first 4 kilometers, so the fare remains a fixed 12 pesos.
- **Case 2:** When \( d > 4 \), we calculate the fare by adding 1 peso for each additional kilometer beyond 4 kilometers. The term \( (d - 4) \) represents the number of kilometers beyond the initial 4, and adding this to the base fare of 12 gives the total fare.

### Simplified Piecewise Function:
For \( d > 4 \), we can simplify the fare calculation:

\[
F(d) = d + 8 \quad \text{if } d > 4.
\]

So the complete piecewise function can also be written as:

\[
F(d) = 
\begin{cases} 
12 & \text{if } 0 \leq d \leq 4, \\
d + 8 & \text{if } d > 4.
\end{cases}
\]

This function now provides the fare based on the distance \( d \).

Would you like more details, or do you have any questions?

Here are some related questions you might consider:
1. What is the fare if Leona travels 3 kilometers?
2. What is the fare if Leona travels 5 kilometers?
3. How would the function change if the fare increased by 2 pesos for each kilometer beyond the first 4?
4. How can we represent the fare in a different mathematical form, like a step function?
5. What would the fare be if the initial fixed distance was 6 kilometers instead of 4?

**Tip:** Understanding piecewise functions is essential when modeling situations with different conditions or rates, such as transportation fares, tax brackets, or material costs.

Leona will go to a mall which is & kilometers from her house and the bus fare costs 12 pesos for the first 4 kilometers and each additional integer kilometer adds 1 peso to the fare. Use a piecewise function to represent the bus fare in terms of the distance (d) in kilometers.

Learn how to calculate the bus fare for a trip using a piecewise function. This problem explains how to represent a bus fare that costs 12 pesos for the first 4 kilometers and adds 1 peso for each additional kilometer. Suitable for students in Grades 8-10.

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