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The problem asks for a **piecewise definition** of the monthly charge for electricity in the winter months. Here’s how we can break it down:

From the image, we see that the electricity rate table for the **winter months** (November to June) is structured as follows:

- **Base charge**: \$6.50
- **First 300 kWh or less**: \$0.0550 per kWh
- **Over 300 kWh**: \$0.0800 per kWh

Now, for the piecewise function $ W(x) $, where $ x $ is the number of kilowatt-hours (kWh) used:

### For part (a):
We are asked to write a piecewise definition of the monthly charge.

The base charge is applied in both cases, so we will add that to each piece of the function. The charge for different values of $ x $ is as follows:

1. **If $ 0 \leq x \leq 300 $:**  
   The charge is \$6.50 (base charge) plus $ 0.0550 \times x $ for the energy used.

   So for this range, the charge function is:  
   \[
   W(x) = 6.50 + 0.0550x
   \]

2. **If $ x > 300 $:**  
   For the first 300 kWh, the rate is \$0.0550 per kWh, and for the energy above 300 kWh, the rate is \$0.0800 per kWh. Therefore, the charge will be:
   \[
   W(x) = 6.50 + (0.0550 \times 300) + 0.0800(x - 300)
   \]
   Simplifying the part $ (0.0550 \times 300) $, which equals \$16.50:
   \[
   W(x) = 6.50 + 16.50 + 0.0800(x - 300)
   \]
   \[
   W(x) = 23.00 + 0.0800(x - 300)
   \]

### Final piecewise function:
\[
W(x) =
\begin{cases}
6.50 + 0.0550x & \text{if } 0 \leq x \leq 300 \\
23.00 + 0.0800(x - 300) & \text{if } x > 300
\end{cases}
\]

### For part (b):
The graph of $ W(x) $ will be piecewise linear:

- The first piece, for $ 0 \leq x \leq 300 $, is a straight line with a slope of 0.0550 and a y-intercept of 6.50.
- The second piece, for $ x > 300 $, starts at $ x = 300 $ with a value of \$23.00 and continues with a slope of 0.0800.

Would you like me to graph this piecewise function or explain any part further?

### 5 Related Questions:
1. How do you calculate the slope of each segment in a piecewise function?
2. What are the key steps for graphing a piecewise linear function?
3. How would the piecewise function change for summer months if different rates apply?
4. Can you calculate the total charge for using 400 kWh?
5. How does the base charge affect the total cost in the piecewise function?

### Tip:
When simplifying piecewise functions, always calculate the constants before moving to the variable terms. This ensures accuracy in your final expression.

Write a piecewise definition of the monthly charge W(x) for a customer who uses x kWh in a winter month. Determine the graph of W(x).

This problem covers how to write a piecewise function for electricity charges based on usage in winter months. It explores a base charge and variable rates for different energy consumption levels, demonstrating how to create and graph the function.

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