Construct a piecewice function based on the problem. Show your solution and write it in a 1 whole sheet of paper.

Driving lessons require a rental car fee of P 500.00 for the first 8 km. and for every kilometer added charges an additional fee of P 50.00. Write a piecewise function d for the problem and find the total rental fee if the car travelled as follows:

A6km

10km

c. 15.5km

D. 20.2km

We are given the problem about a rental car fee structure, which can be described with a piecewise function. Let’s break it down step by step and construct the piecewise function accordingly.

### Problem Breakdown:
- **Base fee**: The rental car fee is **P 500** for the first 8 kilometers.
- **Additional fee**: After 8 kilometers, there’s an additional charge of **P 50 per kilometer**.

Let’s define the **piecewise function** \( d(x) \), where \( x \) is the number of kilometers traveled.

### Step 1: Construct the Piecewise Function
- If the car travels **8 kilometers or less**: the total fee is a fixed amount of **P 500**.
- If the car travels **more than 8 kilometers**: the total fee is **P 500 + P 50 for each kilometer exceeding 8 kilometers**.

Thus, the **piecewise function** \( d(x) \) can be written as:

\[
d(x) =
\begin{cases}
500, & \text{if } 0 \leq x \leq 8 \\
500 + 50(x - 8), & \text{if } x > 8
\end{cases}
\]

Where:
- \( x \) represents the total kilometers traveled.
- \( d(x) \) is the total rental fee.

### Step 2: Calculate the Total Rental Fee for Different Distances
We now use the piecewise function to find the total rental fee for different distances traveled.

#### (a) \( x = 6 \) km:
Since 6 km is less than or equal to 8 km, we use the first part of the piecewise function:

\[
d(6) = 500
\]

So, the total rental fee for 6 km is **P 500**.

#### (b) \( x = 10 \) km:
Since 10 km is greater than 8 km, we use the second part of the piecewise function:

\[
d(10) = 500 + 50(10 - 8) = 500 + 50(2) = 500 + 100 = 600
\]

So, the total rental fee for 10 km is **P 600**.

#### (c) \( x = 15.5 \) km:
Since 15.5 km is greater than 8 km, we use the second part of the piecewise function:

\[
d(15.5) = 500 + 50(15.5 - 8) = 500 + 50(7.5) = 500 + 375 = 875
\]

So, the total rental fee for 15.5 km is **P 875**.

#### (d) \( x = 20.2 \) km:
Since 20.2 km is greater than 8 km, we use the second part of the piecewise function:

\[
d(20.2) = 500 + 50(20.2 - 8) = 500 + 50(12.2) = 500 + 610 = 1110
\]

So, the total rental fee for 20.2 km is **P 1110**.

---

### Final Piecewise Function and Results Summary:
The piecewise function \( d(x) \) is:

\[
d(x) =
\begin{cases}
500, & \text{if } 0 \leq x \leq 8 \\
500 + 50(x - 8), & \text{if } x > 8
\end{cases}
\]

The total rental fees for the given distances are:
- \( d(6) = P 500 \)
- \( d(10) = P 600 \)
- \( d(15.5) = P 875 \)
- \( d(20.2) = P 1110 \)

---

Would you like further details or have any questions?

Here are some related questions you could explore:

1. What would the rental fee be for 12 km?
2. How would the rental fee change if the rate after 8 km increased to P 60 per km?
3. Can you write a general formula for calculating the rental fee for any distance?
4. What is the domain and range of the piecewise function?
5. How does the piecewise function change if the base fee is reduced to P 400 for the first 8 km?

**Tip**: When constructing piecewise functions, carefully examine where different conditions apply based on intervals or thresholds.

Learn how to construct a piecewise function for calculating rental car fees based on distance traveled. This example covers the concept of piecewise functions in mathematics, applied to real-world scenarios. Suitable for high school students and beyond.

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