Question

A Norman window has the shape of a rectangle capped by a semicircle. What is the length of the base of a Norman window of maximum area if the perimeter of the window equals 7?

Answer 1

Let r = radius of semi-circle (half the base).
Let h = height of rectangle

Perimeter = 9 = 2r + 2h + pi r [base + sides + arc]

r (pi + 2) + 2h = 9
2h = 9 - r (pi + 2)
h = 9/2 - r/2 (pi + 2)

Area = 2r * h + pi r^2 / 2 =
2r (9/2 - r/2 (pi + 2)) + pi r^2 / 2=
9 r - r^2 (pi + 2) + pi r^2 / 2 =
9 r - r^2 (pi + 2 - p/2) =
9 r - r^2 (pi/2 + 2)

Derivative is:
9 - r (pi + 4)
Setting that to 0 to find maximum:
r = 9 / (pi + 4) = 9 / (4 + pi) = 1.2602231

For this value, call it 1.26,
arc = pi * 1.26 = 3.9584
height = ( 9 - 2.52 - 3.96 ) / 2 = 1.26

So the rectangular part consists of 2 squares 1.26 x 1.26
The base is 2.52 (or 18 / (4 + pi) to be exact).

Maximum area problems of this kind *always* turn out
to have the dimensions "most evenly divided" into parts of
the perimeter (square, circle, other similar shape).

Answer 2

http://answers.yahoo.com/question/index?qid=20090326130700AAXBeYU

Answer 3

http://answers.yahoo.com/question/index?qid=20101102082151AAmYcFJ

Answer 4

I was looking at those problems and some of them I don't understand where they get of the number so I was completely lost.

Answer 5

ok

Answer 6

so you know what a norman window is?