A Norman window has the shape of a rectangle surmounted by a

A Norman window has the shape of a rectangle surmounted by a semicircle, as shown in the figure below. A Norman window with perimeter 60 ft is to be constructed. Find a function that models the area, A, of the window in terms of width x. A(x) =

Solution

Let length be y

rdius , r= x/2

So, perimeter : x+2y +pi*radius

= x+ 2y +pix/2

x+ 2y +pix/2 = 60

2y = 60 -pix/2 -x

y = 30 - pix/4 -x/2 = 30 - x( pi/4 + 1/2)

= 30 -x(2pi+1)/4

Area = length*width + semicircle

= x*(30 -x(2pi+1)/4) + pix^2/4

= 30x - x^2(2pi +1)/4 +pix^2/4

= 30 x + pix^2(2/4 +1/4) -x^2/4

= 30x + 3pix^2/4 - x^2/4

= 30x + x^2/4(3pi -1)