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The problem involves a rectangular window topped with a semicircle. Let's break it down step by step:

### Given:
- The window consists of a rectangle and a semicircle.
- The height of the rectangular part is given as \( 1 + 3w \), where \( w \) is the width.
- You need to find the total area \( A(w) \) of the window in terms of \( w \), which is the width.

### Step 1: Area of the rectangular part
The area of a rectangle is given by:
\[
\text{Area of rectangle} = \text{width} \times \text{height}
\]
Here, the width is \( w \) and the height is \( 1 + 3w \). So the area of the rectangular part is:
\[
A_{\text{rectangle}} = w \times (1 + 3w)
\]

### Step 2: Area of the semicircular part
The area of a full circle is given by \( \pi r^2 \), where \( r \) is the radius. Since the semicircle is half of a circle, the area is:
\[
A_{\text{semicircle}} = \frac{1}{2} \pi r^2
\]
Here, the radius \( r \) is \( \frac{w}{2} \) (since the semicircle sits on top of the rectangle, and its diameter is equal to the width \( w \) of the rectangle). So the area of the semicircle is:
\[
A_{\text{semicircle}} = \frac{1}{2} \pi \left( \frac{w}{2} \right)^2 = \frac{\pi w^2}{8}
\]

### Step 3: Total Area
The total area \( A(w) \) of the window is the sum of the area of the rectangle and the area of the semicircle:
\[
A(w) = A_{\text{rectangle}} + A_{\text{semicircle}} = w(1 + 3w) + \frac{\pi w^2}{8}
\]

### Answer Choices:
Looking at the answer choices, none of them have the fraction \( \frac{\pi w^2}{8} \), but they approximate the semicircle's area by dividing it by 2 directly without the \( 8 \) factor. Therefore, the closest option should match the general form of:
\[
A(w) = w(1 + 3w) + \frac{\pi w^2}{2}
\]
which corresponds to **option C**.

Would you like more details on any part of the solution? Here are 5 additional questions to deepen your understanding:

1. How does the formula for the area of a semicircle change if the radius changes?
2. Can you calculate the area if the width of the window is 4 meters?
3. How would the total area change if the height of the rectangle were instead \( 2 + 3w \)?
4. Why is the radius of the semicircle \( \frac{w}{2} \)?
5. What would happen if the top of the window were an ellipse instead of a semicircle?

**Tip**: When dealing with shapes that combine multiple geometric forms, always treat each section separately to simplify calculations.

A rectangular window is topped with a semicircle. The height of the rectangular part is 1 more than 3 times its width, w meters. Which function represents the total area, A, of the window in terms of the width?

Learn how to calculate the total area of a window composed of a rectangle topped with a semicircle. The height of the rectangular part is 1 more than 3 times its width. This problem covers key geometric and algebraic concepts and is suitable for students in Grades 9-12.

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