SOLUTION: " A Norman window is a rectangle with a semicircle on top. Big Sky window is designing a Norman window that will require 24 feet of trim. What dimensions will allow the maximum amo

Algebra ->  Customizable Word Problem Solvers  -> Geometry -> SOLUTION: " A Norman window is a rectangle with a semicircle on top. Big Sky window is designing a Norman window that will require 24 feet of trim. What dimensions will allow the maximum amo      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 327684: " A Norman window is a rectangle with a semicircle on top. Big Sky window is designing a Norman window that will require 24 feet of trim. What dimensions will allow the maximum amount of light to enter the house?"
I'm really lost on this one. Thank you for your time.
Answer by galactus(183) About Me  (Show Source):
You can put this solution on YOUR website!
Let the height of the rectangular part be y and the radius of the semi-circular part be r. Therefore, the perimeter of the window is
S=pi%2Ar%2B2y%2B2r=24
y=%2824-2r-pi%2Ar%29%2F2
The area is A=pi%2Ar%5E2%2F2%2B2ry
Sub y into A and it simplifies down to:
A=-pi%2Ar%5E2%2F2-2r%5E2%2B24r=-%28pi%2F2%2B2%29r%5E2%2B24r
This is what must be maximized to allow the most light in.
It can be done with or without calculus. You did not specify.
To find the max without calc, we can use the formula for the vertex of a parabola, r=-b%2F%282a%29
Using this we find r=24%2F%28pi%2B4%29
This can be subbed into the y equation above to find the y dimension and thus the area needed to maximize light entry.