Problem Breakdown and Piecewise Function

Given:

• Rental car fee for the first 8 km: P 500.00
• Additional fee per kilometer beyond 8 km: P 50.00

We need to create a piecewise function that represents the total rental fee, $d(x)$, where $x$ is the total distance traveled in kilometers.

Constructing the Piecewise Function

1. For distances $x \leq 8$ km:

   • The rental fee is constant at P 500.00.

2. For distances $x > 8$ km:

   • The rental fee is P 500.00 for the first 8 km, plus an additional P 50.00 for every kilometer beyond 8 km.

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

\begin{cases} 500, & \text{if } 0 \leq x \leq 8 \\ 500 + 50(x - 8), & \text{if } x > 8 \end{cases}$$ ### Calculating the Rental Fee for Different Distances 1. **For $$x = 6$$ km:** - Since $$6 \leq 8$$, we use the first part of the piecewise function: $$d(6) = 500$$ **Rental Fee: P 500.00** 2. **For $$x = 10$$ km:** - Since $$10 > 8$$, we use the second part of the piecewise function: $$d(10) = 500 + 50(10 - 8) = 500 + 50(2) = 500 + 100 = 600$$ **Rental Fee: P 600.00** 3. **For $$x = 15.5$$ km:** - Since $$15.5 > 8$$, we use the second part of the piecewise function: $$d(15.5) = 500 + 50(15.5 - 8) = 500 + 50(7.5) = 500 + 375 = 875$$ **Rental Fee: P 875.00** 4. **For $$x = 20.2$$ km:** - Since $$20.2 > 8$$, we use the second part of the piecewise function: $$d(20.2) = 500 + 50(20.2 - 8) = 500 + 50(12.2) = 500 + 610 = 1110$$ **Rental Fee: P 1110.00** ### Summary of Results - **A. For 6 km:** Rental Fee = **P 500.00** - **B. For 10 km:** Rental Fee = **P 600.00** - **C. For 15.5 km:** Rental Fee = **P 875.00** - **D. For 20.2 km:** Rental Fee = **P 1110.00** ### Writing it on a Whole Sheet of Paper You can write this solution in a clear, step-by-step manner on a whole sheet of paper as follows: 1. Start by stating the problem and the given values. 2. Define the piecewise function. 3. Calculate the rental fee for each specific distance. 4. Summarize the results. Would you like me to simulate how this would appear on a whole sheet of paper, or would you like more details on any part?