To find the dimensions of a Norman window of maximum area, we need to maximize the area of the window while considering the given perimeter constraint.
Let's denote the width of the rectangular window as 2x and the height as y.
The perimeter of the rectangular window is 2x + 2y.
Since the total perimeter is given as 16 feet, we have the equation:
2x + 2y + πx = 16
Simplifying this equation, we get:
2x + 2x + πx = 16
Combining like terms, we have:
4x + πx = 16
Factoring out x, we get:
x(4 + π) = 16
Dividing both sides by (4 + π), we obtain:
x = 16 / (4 + π)
Now, to find the height y, we can substitute the value of x back into the equation 2x + 2y = 16:
2(16 / (4 + π)) + 2y = 16
Simplifying this equation, we have:
32 / (4 + π) + 2y = 16
Subtracting 32 / (4 + π) from both sides, we get:
2y = 16 - 32 / (4 + π)
Dividing both sides by 2, we obtain:
y = (16 - 32 / (4 + π)) / 2
Therefore, the dimensions of the Norman window that maximizes the area can be calculated as follows:
Width (2x) = 16 / (4 + π) feet
Height (y) = (16 - 32 / (4 + π)) / 2 feet