Worksheet titled "Piecewise Word Problems" featuring three real-world scenarios requiring the creation of piecewise functions to solve for specific values.

Problem 1: Summer Job with Overtime

#### a. Write a piecewise function to represent this problem.
The problem states that you earn:
- $7.00 per hour for the first 40 hours.
- Time-and-a-half (1.5 times the hourly rate) for any hours worked beyond 40 hours.

Let \( h \) be the number of hours worked, and let \( P(h) \) be the total pay.

The piecewise function can be written as:
\[
P(h) =
\begin{cases}
7h & \text{if } 0 \leq h \leq 40 \\
7 \cdot 40 + 1.5 \cdot 7 \cdot (h - 40) & \text{if } h > 40
\end{cases}
\]

Simplifying the second case:
\[
P(h) =
\begin{cases}
7h & \text{if } 0 \leq h \leq 40 \\
280 + 10.5(h - 40) & \text{if } h > 40
\end{cases}
\]

#### b. How much money do you make if you work 45 hours?
Since \( h = 45 \), we use the second case of the piecewise function:
\[
P(45) = 280 + 10.5(45 - 40)
\]
\[
P(45) = 280 + 10.5 \cdot 5
\]
\[
P(45) = 280 + 52.5
\]
\[
P(45) = 332.5
\]

So, the total pay for working 45 hours is:
\[
\boxed{332.5}
\]

---

Problem 2: Cell Phone Plan

#### a. Write a piecewise function to represent this situation.
The cell phone plan costs:
- $75 per month for unlimited talk and 500 text messages.
- An additional $0.10 per text message for any texts sent beyond 500.

Let \( t \) be the number of text messages sent, and let \( C(t) \) be the total cost.

The piecewise function can be written as:
\[
C(t) =
\begin{cases}
75 & \text{if } 0 \leq t \leq 500 \\
75 + 0.10(t - 500) & \text{if } t > 500
\end{cases}
\]

#### b. How much will it cost if you send 750 text messages?
Since \( t = 750 \), we use the second case of the piecewise function:
\[
C(750) = 75 + 0.10(750 - 500)
\]
\[
C(750) = 75 + 0.10 \cdot 250
\]
\[
C(750) = 75 + 25
\]
\[
C(750) = 100
\]

So, the total cost for sending 750 text messages is:
\[
\boxed{100}
\]

---

Problem 3: Buying Snickers

#### Write a piecewise function to represent this situation.
The cost of buying snickers is:
- $3.45 per bag if fewer than 4 bags are bought.
- $3.00 per bag if 4 or more bags are bought.

Let \( n \) be the number of bags of snickers bought, and let \( S(n) \) be the total cost.

The piecewise function can be written as:
\[
S(n) =
\begin{cases}
3.45n & \text{if } 0 \leq n < 4 \\
3.00n & \text{if } n \geq 4
\end{cases}
\]

---

Final Answers:

1. a. The piecewise function for the summer job is:
\[
P(h) =
\begin{cases}
7h & \text{if } 0 \leq h \leq 40 \\
280 + 10.5(h - 40) & \text{if } h > 40
\end{cases}
\]
b. The total pay for working 45 hours is:
\[
\boxed{332.5}
\]

2. a. The piecewise function for the cell phone plan is:
\[
C(t) =
\begin{cases}
75 & \text{if } 0 \leq t \leq 500 \\
75 + 0.10(t - 500) & \text{if } t > 500
\end{cases}
\]
b. The total cost for sending 750 text messages is:
\[
\boxed{100}
\]

3. The piecewise function for buying snickers is:
\[
S(n) =
\begin{cases}
3.45n & \text{if } 0 \leq n < 4 \\
3.00n & \text{if } n \geq 4
\end{cases}
\]