Answer: The dimensions of the rectangular part of the Norman window that would allow in as much light as possible, given 12 feet of framing material available, are approximately 4 feet by 8 feet.
Step-by-step explanation:
Let's assume that the height of the rectangular part of the Norman window is "h" and the width is "w". Then the diameter of the semicircle is also "w". The total amount of framing material needed is the sum of the perimeter of the rectangular part and half the circumference of the semicircle:
Perimeter of rectangular part = 2h + 2w
Circumference of semicircle = 1/2πw
Total framing material = 2h + 2w + 1/2πw
We want to maximize the amount of light entering the window, which is proportional to the area of the rectangular part of the window. The area of the rectangular part is given by:
Area of rectangular part = hw
Now we can use the constraint that there is only 12 feet of framing material available:
2h + 2w + 1/2πw = 12
Solving for h in terms of w:
h = (12 - 2w - 1/2πw)/2
Substituting this expression for h into the formula for the area of the rectangular part:
Area of rectangular part = w(12 - 2w - 1/2πw)/2
We can now use calculus to find the value of w that maximizes this area. Taking the derivative of the area with respect to w and setting it equal to zero:
d/dw[w(12 - 2w - 1/2πw)/2] = 0
Simplifying and solving for w:
w = 4π/(4 + π)
Substituting this value of w into the expression for h:
h = (12 - 2w - 1/2πw)/2
h ≈ 8
Therefore, the dimensions of the rectangular part of the Norman window that allow in as much light as possible, given 12 feet of framing material available, are approximately 4 feet by 8 feet.