Piecewise function, A house painter charges $350 per hour for the first 40 hours he works, time and a half for the 10 hours after that, and double time for all hours after that. How much does he earn for a 70-hour week? Represent using a function

To represent the earnings of the painter using a piecewise function based on the number of hours worked, we need to account for the different rates for each segment of hours.

1. For the first 40 hours, the painter charges $350 per hour.
2. For the next 10 hours (i.e., from 41 to 50 hours), he charges "time and a half," which is $ 1.5 \times 350 = 525 $ dollars per hour.
3. For any hours beyond 50 (i.e., 51 hours and beyond), he charges double time, which is $ 2 \times 350 = 700 $ dollars per hour.

### Define the function:

Let $ h $ be the total number of hours worked, and $ E(h) $ be the earnings.

\[
E(h) = \begin{cases} 
350h & \text{if } 0 \leq h \leq 40 \\
350 \times 40 + 525 \times (h - 40) & \text{if } 40 < h \leq 50 \\
350 \times 40 + 525 \times 10 + 700 \times (h - 50) & \text{if } h > 50
\end{cases}
\]

### Earnings Calculation for a 70-hour week:

Since 70 hours is more than 50, we use the third part of the piecewise function:

\[
E(70) = 350 \times 40 + 525 \times 10 + 700 \times (70 - 50)
\]

Now, let's compute this.

1. First 40 hours: $ 350 \times 40 = 14,000 $
2. Next 10 hours: $ 525 \times 10 = 5,250 $
3. Remaining 20 hours: $ 700 \times 20 = 14,000 $

Now, sum up the earnings:

\[
E(70) = 14,000 + 5,250 + 14,000 = 33,250
\]

### Conclusion:
The painter earns **$33,250** for a 70-hour week.

Would you like a more detailed explanation or have any further questions?

Here are 5 related questions to expand this:

1. How would the function change if the painter also offered a discount after 80 hours?
2. What if the rates for overtime were different, like 2x for all hours beyond 40?
3. How much would the painter earn for a 55-hour week?
4. Can this piecewise function be plotted to show earnings based on hours worked?
5. What if the painter also included a flat fee for setup time, regardless of hours worked?

**Tip:** Piecewise functions are often used to model real-world problems with varying conditions, like different pay rates or tax brackets.

A house painter charges $350 per hour for the first 40 hours he works, time and a half for the 10 hours after that, and double time for all hours after that. How much does he earn for a 70-hour week? Represent using a function.

Explore how to calculate a painter's earnings for a 70-hour work week using piecewise functions. Understand the different hourly rates: $350 for the first 40 hours, time and a half for the next 10 hours, and double time for any additional hours. Suitable for students in Grades 10-12 learning about linear equations and piecewise functions.

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