SOLUTION: A Norman window is a window with a semi-circle on top of regular rectangular window. (See the picture.) What should be the dimensions of the rectangular part of a Norman window to

Algebra ->  Surface-area -> SOLUTION: A Norman window is a window with a semi-circle on top of regular rectangular window. (See the picture.) What should be the dimensions of the rectangular part of a Norman window to       Log On


   

Question 1157160: A Norman window is a window with a semi-circle on top of regular rectangular window. (See the picture.) What should be the dimensions of the rectangular part of a Norman window to allow in as much light as possible, if there are only 12 ft of the frame material available?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
x, the diameter and dimension of the one side with the semicircle
w, the other rectangular dimension

Area A:
A=%281%2F2%29pi%2A%28x%2F2%29%5E2%2Bxw
highlight_green%28A=pi%2Ax%2F8%2Bxw%29

Amount of frame material, 12 feet:
%281%2F2%292%2Api%2A%28x%2F2%29%2Bx%2B2w=12
highlight_green%28pi%2Ax%2F2%2Bx%2B2w=12%29

The two equations are just after starting, and not the finished solution ,...

Answer by ikleyn(52887) About Me  (Show Source):
You can put this solution on YOUR website!
.

Let x = width and diameter;

    y = height of the rectangle part.


Then the perimeter   

P = x+%2B+2y+%2B+%28pi%2Ax%29%2F2%29  ====>  x+%2B+%28pi%2Ax%29%2F2 + 2y = 12  ====>  y = 6+-+x%2F2+-+%28pi%2Ax%29%2F4.


The area A = xy + %281%2F2%29%2Api%2A%28x%2F2%29%5E2 = x%2A%286-x%2F2+-+%28pi%2Ax%29%2F4%29 + %28pi%2F2%29%2A%28x%2F2%29%5E2 = 6x - x%5E2%2F2 - %28pi%2F4%29%2Ax%5E2 + %28pi%2F8%29%2Ax%5E2 = -x%5E2%2F2 + 6x - %28pi%2F8%29%2Ax%5E2



Then  the condition for the maximum area  %28dA%29%2F%28dx%29 = 0  takes the form


-x+%2B+6+-+%28pi%2F4%29%2Ax = 0,   or   x%2A%281%2Bpi%2F4%29 = 6  ====> x = 6%2F%281%2Bpi%2F4%29 = 6%2F%281+%2B+%283.14%2F4%29%29 = 3.36 ft.


Answer.  The maximum area is at x = 3.36 ft  (the width), 

         y = 6+-+x%2F2+-+%28pi%2Ax%29%2F4 = 6+-+3.36%2F2+-+%283.14%2A3.36%29%2F4 = 1.68 ft (the height).

Solved.