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Let's break down the problem step by step.

### (a) Write a function for weekly pay $ W $:
The construction worker earns:
- $17 per hour for the first 40 hours.
- $25.50 per hour for any hours worked beyond 40 hours.

Let $ h $ be the total number of hours worked in a week.

The function for the weekly pay $ W(h) $ would be:
- If $ h \leq 40 $, then $ W(h) = 17h $ (since no overtime is worked).
- If $ h > 40 $, then $ W(h) = 17 \times 40 + 25.5 \times (h - 40) $.

So, the function can be written as:
\[
W(h) =
\begin{cases}
17h & \text{if } h \leq 40 \\
17 \times 40 + 25.5 \times (h - 40) & \text{if } h > 40
\end{cases}
\]

### (b) She earned $896.75 in one week. How much overtime did she work?

To solve this, we know her total earnings, so we can set up the equation using the function $ W(h) $ and solve for $ h $.

If she worked overtime, the equation will be:
\[
896.75 = 17 \times 40 + 25.5 \times (h - 40)
\]
Let me calculate the value of $ h $ from this equation.

The total number of hours worked was 48.5 hours.

Since she worked 48.5 hours and the first 40 hours are regular hours, the overtime hours worked are:
\[
48.5 - 40 = 8.5 \text{ hours}.
\]

Thus, she worked **8.5 hours** of overtime.

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Do you want further details on this, or have any questions? Here are 5 related questions:

1. How would the function change if the overtime pay rate increased to $30 per hour?
2. What would her weekly pay be if she worked 50 hours in a week?
3. How much would she earn for 35 hours of work?
4. What is the overtime rate as a percentage of her regular pay rate?
5. If she worked 60 hours, how much would be earned from overtime alone?

**Tip**: When working with piecewise functions, clearly define the conditions and calculate each part separately for accuracy.

A construction worker earned $17 per hour for the first 40 hours of work and $25.50 per hour for work in excess of 40 hours. (a) Write a function to model her weekly pay W in terms of the number of hours she worked that week. (b) One week she earned $896.75. How much overtime did she work?

Learn how to create a piecewise function to model a construction worker's weekly pay based on regular and overtime hours. The problem involves calculating pay for the first 40 hours at $17 per hour and $25.50 per hour for overtime. Suitable for students in Grades 10-12.

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