A Norman window is a rectangle with a semicircle on top. Suppose that the perimeter of a particular Norman window is to be 56 ft. What should its dimensions be in order to allow the maximum amount of light to enter through the window?

Dimensions of the window in order to allow maximum light is and

Step-by-step explanation:

Consider following Norman window, assuming ABCD as rectangle and arc AD as semicircle with center at E and radius r. (Refer attachment)

Given that perimeter of window is 56 ft. Therefore perimeter of window is given as,  

Calculate arc AD as follows,  

Let, x denote radius of semi circle. That is, r=x  

Since AD is the diameter of semi circle  

So AD = 2 r = 2 x.  

Now perimeter of semicircle is equal to circumference of semicircle, so calculate circumference of semicircle as follows  

Circumference of circle is . So half of it will be  

So,

Calculate AB , BC and AD as follows,  

Consider rectangle ABCD,  

Since, AD is the diameter of semi circle which is also one of the side of the rectangle.  

So AD = 2 r = 2 x.  

Since AD is parallel to BC. Therefore, AD=BC=2x. Also length of rectangle be y  

Substituting the value,  

Perimeter = AB + BC + CD + arc AD  

To calculate value of y,subtracting 2x and on both sides,  

Dividing by 2,

Now calculate area of window.  

Area = Area of rectangle + Area of semicircle

From diagram,

Substituting value of y in above equation,

Simplifying,  

In order to find the maximum of area function, differentiate the equation with respect to x and find the critical points.  

Applying difference rule of derivative,  

Applying constant multiple rule of derivative,  

Applying power rule of derivative,

Now find the critical number by solving as follows,

Since there is only one critical point, directly substitute the value of x into equation of A. If value of A is greater than 0, then the area is maximum at critical point.  

Calculating the above expression,  

So area is greater than 0.  

Now calculate value of y,  

Hence dimensions are and