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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.

A Norman window has the shape of a rectangle surmounted by a semicircle of diameter equal to the width of the rectangle. If the perimeter of the window is 12 feet, what dimensions will admit the most light (maximize the area)?

Explore how to maximize the area of a Norman window with a fixed perimeter of 12 feet. Using optimization and geometry, solve for the dimensions that allow the most light. Suitable for students in Grades 10-12.

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