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To solve the given problem involving the Norman window, let's break it down step-by-step.

### **1. Understanding the components:**
- The window consists of:
  1. A **semicircle** at the top with **radius \( r \)**.
  2. A **rectangle** below the semicircle, with:
     - **Width \( 2r \)** (since the width of the rectangle equals the diameter of the semicircle).
     - **Height \( w \)** (denoted in the figure).

### **2. Perimeter constraint:**
The **perimeter** of the Norman window includes:
- The **semicircle's curved edge**: \( \pi r \) (half of the circumference of a full circle).
- The **two vertical sides** of the rectangle: \( 2w \).
- The **bottom side** of the rectangle: \( 2r \) (same as the diameter of the semicircle).

Thus, the **total perimeter** is:
\[
\pi r + 2w + 2r = 23 \, \text{ft}.
\]

### **3. Optimization goal:**
We need to **maximize the area** to allow the most light. The total area consists of:
- **Area of the semicircle**: \( \frac{1}{2} \pi r^2 \).
- **Area of the rectangle**: \( 2r \cdot w \).

Thus, the **total area** is:
\[
A(r, w) = \frac{1}{2} \pi r^2 + 2r w.
\]

### **4. Solve for \( w \) in terms of \( r \) using the perimeter:**
From the perimeter equation:
\[
\pi r + 2w + 2r = 23.
\]
Rearrange to solve for \( w \):
\[
2w = 23 - \pi r - 2r,
\]
\[
w = \frac{23 - \pi r - 2r}{2}.
\]

### **5. Express the area in terms of \( r \) only:**
Substitute the expression for \( w \) into the area formula:
\[
A(r) = \frac{1}{2} \pi r^2 + 2r \left( \frac{23 - \pi r - 2r}{2} \right).
\]

Simplify:
\[
A(r) = \frac{1}{2} \pi r^2 + r(23 - \pi r - 2r).
\]
\[
A(r) = \frac{1}{2} \pi r^2 + 23r - \pi r^2 - 2r^2.
\]
\[
A(r) = -\frac{1}{2} \pi r^2 - 2r^2 + 23r.
\]

### **6. Maximize the area:**
To find the value of \( r \) that maximizes the area, take the derivative of \( A(r) \) with respect to \( r \) and set it equal to zero:
\[
\frac{dA}{dr} = -\pi r - 4r + 23 = 0.
\]
\[
-(\pi + 4)r + 23 = 0.
\]
\[
(\pi + 4)r = 23.
\]
\[
r = \frac{23}{\pi + 4}.
\]

### **7. Calculate the corresponding \( w \):**
Now that we have \( r \), substitute it into the equation for \( w \):
\[
w = \frac{23 - \pi r - 2r}{2}.
\]

Using \( r = \frac{23}{\pi + 4} \), you can substitute this value into the expression to compute \( w \).

---

Would you like me to proceed with the numerical calculation for these values or clarify any step? 

### 5 Related Questions:
1. How do we ensure this critical point yields the maximum area?
2. What is the area of the Norman window with these dimensions?
3. How does the perimeter constraint limit the design?
4. What happens if the perimeter was increased or decreased?
5. Can calculus guarantee the most optimal dimensions in all cases?

### Tip:
Always verify your derivative calculations to ensure you correctly identify critical points for optimization problems.

A Norman window consists of a semicircle adjoining a rectangular window. If the perimeter of the Norman window is 23 ft, find the dimensions of r (the radius of the semicircle) and w (the height of the rectangle) that allow the window to admit the most light.

Learn how to optimize the dimensions of a Norman window, combining a semicircle and rectangle, to admit the most light. Explore the geometric and calculus-based approach, solving for r and w under the constraint of a 23 ft perimeter. Suitable for advanced high school or early college students.

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