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

To construct the piecewise function based on the given problem and solve for the total rental fee for each distance, let's break down the problem step by step.

### Problem Breakdown:
1. **Rental Car Fee**:
   - The base fee for the first 8 km is \( \text{P}500.00 \).
   - For any distance beyond 8 km, an additional fee of \( \text{P}50.00 \) per kilometer is charged.

### Piecewise Function Definition:
Let \( d(x) \) be the total rental fee based on the distance \( x \) in kilometers.

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

- The first case covers distances from 0 to 8 km, where the fee is a flat rate of \( \text{P}500.00 \).
- The second case covers distances greater than 8 km, where the fee is \( \text{P}500.00 \) plus an additional \( \text{P}50.00 \) for every kilometer beyond 8 km.

### Solutions for the Given Distances:

1. **For 6 km**:
   Since \( 6 \leq 8 \), use the first part of the piecewise function:
   \[
   d(6) = 500
   \]
   **Total Fee: P 500.00**

2. **For 10 km**:
   Since \( 10 > 8 \), use the second part of the piecewise function:
   \[
   d(10) = 500 + 50(10 - 8) = 500 + 50(2) = 500 + 100 = 600
   \]
   **Total Fee: P 600.00**

3. **For 15.5 km**:
   Since \( 15.5 > 8 \), use the second part of the piecewise function:
   \[
   d(15.5) = 500 + 50(15.5 - 8) = 500 + 50(7.5) = 500 + 375 = 875
   \]
   **Total Fee: P 875.00**

4. **For 20.2 km**:
   Since \( 20.2 > 8 \), use the second part of the piecewise function:
   \[
   d(20.2) = 500 + 50(20.2 - 8) = 500 + 50(12.2) = 500 + 610 = 1110
   \]
   **Total Fee: P 1110.00**

### Writing It on a Whole Sheet of Paper:
To write this solution neatly on a whole sheet of paper, you should:
1. Write the problem statement at the top.
2. Define the piecewise function clearly with each case explained.
3. Calculate the total rental fees for the given distances, showing all steps.
4. Summarize the results in a table if necessary for clarity.

### Summary of Results:
| Distance (km) | Total Fee (P) |
|---------------|---------------|
| 6.0           | 500.00        |
| 10.0          | 600.00        |
| 15.5          | 875.00        |
| 20.2          | 1110.00       |

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Would you like more details on any part of the solution? Here are some related questions to consider:

1. What would the fee be for traveling exactly 8 km?
2. How would the piecewise function change if the base fee was for the first 10 km instead of 8 km?
3. What if the additional fee per kilometer was P 60.00 instead of P 50.00?
4. How can you represent this piecewise function graphically?
5. What if there was a cap on the maximum fee charged, say P 1200.00, how would that affect the function?

**Tip:** When constructing piecewise functions, always carefully identify the conditions that separate each piece to avoid overlap or gaps in your function definition.

Learn how to construct a piecewise function for calculating rental car fees based on distance travelled. Includes step-by-step solutions for distances of 6 km, 10 km, 15.5 km, and 20.2 km. Suitable for high school students learning piecewise functions in mathematics.

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