A window is in the form of a rectangle, together with a semi-circle on its top side as diameter. If the perimeter of the window is 80 inches, determine the dimensions of the rectangle, so that a maximum amount of light may be admitted.

Answer

To solve the problem of maximizing the area of a window in the form of a rectangle with a semicircle on top, we will follow these steps:

### Step 1: Define Variables
Let:
- \( a \) = breadth of the rectangle
- \( b \) = length of the rectangle
- The radius of the semicircle = \( \frac{b}{2} \)

### Step 2: Write the Perimeter Equation
The perimeter of the window consists of the two sides of the rectangle, the bottom side, and the semicircular arc on top. The perimeter \( P \) can be expressed as:
\[ P = 2a + b + \frac{1}{2} \pi \left(\frac{b}{2}\right) = 80 \]
This simplifies to:
\[ P = 2a + b + \frac{\pi b}{4} = 80 \]

### Step 3: Rearrange for \( a \)
Rearranging the perimeter equation to express \( a \) in terms of \( b \):
\[ 2a = 80 - b - \frac{\pi b}{4} \]
\[ a = 40 - \frac{b}{2} - \frac{\pi b}{8} \]

### Step 4: Write the Area Equation
The area \( A \) of the window can be expressed as the sum of the area of the rectangle and the area of the semicircle:
\[ A = ab + \frac{1}{2} \pi \left(\frac{b}{2}\right)^2 \]
Substituting \( a \):
\[ A = \left(40 - \frac{b}{2} - \frac{\pi b}{8}\right)b + \frac{1}{2} \pi \left(\frac{b^2}{4}\right) \]
This simplifies to:
\[ A = 40b - \frac{b^2}{2} - \frac{\pi b^2}{8} + \frac{\pi b^2}{8} = 40b - \frac{b^2}{2} \]

### Step 5: Differentiate the Area Function
To find the maximum area, we differentiate \( A \) with respect to \( b \):
\[ \frac{dA}{db} = 40 - b \]
Setting the derivative equal to zero for maximization:
\[ 40 - b = 0 \implies b = 40 \]

### Step 6: Find \( a \) Using \( b \)
Substituting \( b = 40 \) back into the equation for \( a \):
\[ a = 40 - \frac{40}{2} - \frac{\pi \cdot 40}{8} \]
\[ a = 40 - 20 - 5\pi \]
Thus, the dimensions of the rectangle are:
- Length \( b = 40 \)
- Breadth \( a = 20 - 5\pi \)

### Final Dimensions
The dimensions of the rectangle that maximize the light admitted through the window are:
- Length \( b = 40 \) inches
- Breadth \( a = 20 - 5\pi \) inches