A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30ft, express the area A of the window as a function of the width x of the window.

To express the area \( A \) of the Norman window as a function of its width \( x \), let's break down the problem step by step.

### Step 1: Define the Dimensions
- Let \( x \) be the width of the rectangle (which is also the diameter of the semicircle).
- Let \( h \) be the height of the rectangle.

### Step 2: Express the Perimeter
The perimeter \( P \) of the Norman window is the sum of the perimeter of the rectangle (excluding the top side, which is replaced by the semicircle) and the perimeter of the semicircle.

- The perimeter of the rectangle excluding the top side is \( 2h + x \) (two vertical sides and the bottom side).
- The perimeter of the semicircle is half the circumference of a full circle, which is \( \frac{1}{2} \times \pi \times x = \frac{\pi x}{2} \).

Thus, the total perimeter is:
\[
P = 2h + x + \frac{\pi x}{2}
\]
Given that the perimeter is 30 ft, we have:
\[
2h + x + \frac{\pi x}{2} = 30
\]

### Step 3: Solve for \( h \)
Rearrange the equation to solve for \( h \):
\[
2h = 30 - x - \frac{\pi x}{2}
\]
\[
h = \frac{30 - x - \frac{\pi x}{2}}{2}
\]

### Step 4: Express the Area
The area \( A \) of the Norman window consists of two parts: the area of the rectangle and the area of the semicircle.

- The area of the rectangle is \( A_{\text{rectangle}} = x \times h \).
- The area of the semicircle is \( A_{\text{semicircle}} = \frac{1}{2} \times \pi \times \left( \frac{x}{2} \right)^2 = \frac{\pi x^2}{8} \).

Thus, the total area \( A \) of the window is:
\[
A = x \times h + \frac{\pi x^2}{8}
\]

### Step 5: Substitute \( h \)
Now, substitute the expression for \( h \) into the area formula:
\[
A = x \times \left( \frac{30 - x - \frac{\pi x}{2}}{2} \right) + \frac{\pi x^2}{8}
\]

This is the area \( A \) of the Norman window as a function of the width \( x \).

### Do you want details or have any questions?

Here are five related questions for further exploration:
1. What is the maximum possible area of the Norman window given the perimeter constraint?
2. How would the area formula change if the perimeter were different?
3. Can you find the dimensions of the window that maximize the area?
4. How does the perimeter constraint impact the shape of the window?
5. How would the problem change if the window were composed of a full circle and a rectangle?

**Tip:** When solving optimization problems, always start by expressing one variable in terms of another using the given constraints.

Learn how to express the area A of a Norman window, which consists of a rectangle surmounted by a semicircle, as a function of its width x. Detailed step-by-step solution provided. Suitable for Grades 9-12.

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