Example 4: Create a Piecewise Function - A construction worker earns $17 per hour for the first 40 hours and $25.50 per hour for overtime. Problem asks to create a function for her paycheck and calculate overtime hours when she earned $896.75.

Problem Analysis:

We are tasked with creating a piecewise function to represent the amount of a construction worker's paycheck based on her hourly earnings. The worker earns:
- $17 per hour for the first 40 hours of work.
- $25.50 per hour for any hours worked beyond 40 hours.

We need to:
1. Create a piecewise function to represent her paycheck.
2. Determine how much overtime she worked if her total earnings for one week were $896.75.

---

Step 1: Create the Piecewise Function

Let \( h \) represent the total number of hours worked in a week. We define the function \( P(h) \) as the total paycheck.

#### Case 1: \( h \leq 40 \)
If the worker works 40 hours or fewer, she earns $17 per hour. Therefore:
\[
P(h) = 17h \quad \text{for} \quad h \leq 40
\]

#### Case 2: \( h > 40 \)
If the worker works more than 40 hours, she earns $17 per hour for the first 40 hours and $25.50 per hour for any additional hours. Let \( h - 40 \) represent the number of overtime hours. Therefore:
\[
P(h) = 17 \cdot 40 + 25.50 \cdot (h - 40) \quad \text{for} \quad h > 40
\]
Simplifying the expression:
\[
P(h) = 680 + 25.50(h - 40)
\]
\[
P(h) = 680 + 25.50h - 1020
\]
\[
P(h) = 25.50h - 340 \quad \text{for} \quad h > 40
\]

#### Final Piecewise Function:
\[
P(h) =
\begin{cases}
17h & \text{if } h \leq 40 \\
25.50h - 340 & \text{if } h > 40
\end{cases}
\]

---

Step 2: Determine Overtime Hours for Total Earnings of $896.75

We are given that the worker earned $896.75 in one week. Since this amount is greater than the maximum possible earnings for 40 hours (\( 17 \times 40 = 680 \)), we know she worked overtime. Therefore, we use the second case of the piecewise function:
\[
P(h) = 25.50h - 340
\]
Set \( P(h) = 896.75 \):
\[
896.75 = 25.50h - 340
\]

#### Solve for \( h \):
Add 340 to both sides:
\[
896.75 + 340 = 25.50h
\]
\[
1236.75 = 25.50h
\]

Divide both sides by 25.50:
\[
h = \frac{1236.75}{25.50}
\]
\[
h = 48.5
\]

#### Calculate Overtime Hours:
The total hours worked is 48.5. Overtime hours are the hours worked beyond 40:
\[
\text{Overtime hours} = h - 40 = 48.5 - 40 = 8.5
\]

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Final Answer:

\[
\boxed{8.5}
\]