This problem describes a "Norman window," which consists of a rectangle surmounted by a semicircle. The objective is to maximize the area of the window while keeping the perimeter constant at 12 feet.

Here’s a step-by-step approach to solve it:

Step 1: Define variables

Let:

• $w$ be the width of the rectangle, which is also the diameter of the semicircle.
• $h$ be the height of the rectangle.
• $r$ be the radius of the semicircle, so $r = \frac{w}{2}$.

Step 2: Write the perimeter equation

The perimeter consists of:

• The perimeter of the semicircle (half the circumference of a full circle), which is $\frac{1}{2} \cdot 2\pi r = \pi r = \pi \frac{w}{2}$.
• The height of the rectangle, counted twice (once for each vertical side), which is $2h$.
• The width of the rectangle $w$.

Thus, the total perimeter is: $w + 2h + \pi \frac{w}{2} = 12$ This simplifies to: $w + 2h + \frac{\pi w}{2} = 12$ This is the equation for the perimeter.

Step 3: Write the area equation

The total area $A$ consists of:

• The area of the rectangle: $A_{\text{rectangle}} = w \cdot h$.
• The area of the semicircle: $A_{\text{semicircle}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi \left( \frac{w}{2} \right)^2 = \frac{\pi w^2}{8}$.

Thus, the total area is: $A = w \cdot h + \frac{\pi w^2}{8}$

Step 4: Solve for $h$

From the perimeter equation: $2h = 12 - w - \frac{\pi w}{2}$ $h = \frac{12 - w - \frac{\pi w}{2}}{2}$

Step 5: Substitute $h$ into the area equation

Substitute this expression for $h$ into the area equation to get a function of $A(w)$. Then, maximize the area by taking the derivative of $A(w)$ with respect to $w$, setting it equal to zero, and solving for $w$.

Would you like further details on finding the specific width and height that maximize the area? Let me know if you have any questions or need more guidance.

Here are 5 follow-up questions that might help you explore this problem further:

1. How do you differentiate the area equation to find the maximum?
2. What is the physical interpretation of the critical points found when solving $A'(w) = 0$?
3. How does the perimeter constraint affect the possible dimensions of the window?
4. What role does the semicircle play in maximizing the window’s area compared to a simple rectangular window?
5. How does changing the perimeter affect the optimal dimensions of the window?

Tip: When solving optimization problems, always ensure that the critical points you find satisfy the given constraints.