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.