SOLUTION: A Norman window has the shape of a rectangle surmounted by a semicircle, as shown in the figure below. A Norman window with perimeter 40 ft is to be constructed.Find the dimensions

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Question 1117967: A Norman window has the shape of a rectangle surmounted by a semicircle, as shown in the figure below. A Norman window with perimeter 40 ft is to be constructed.Find the dimensions of the rectangular portion of the window, such that the full window admits the greatest amount of light. (Round your answers to two decimal places.)
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
(this solution not checked thoroughly)

x, the LENGTH
y, the width
A(x), area xy
Biggest area is if x is also the diameter of the attached semicircle. Radius is x%2F2.

Perimeter becomes after some simplification pi%2Ax%2F2%2Bx%2B2y=40.
From this, y=20-pi%2Ax%2F4-x%2F2.

A%28x%29=xy%2B%281%2F2%29pi%28x%2F2%29%5E2
A%28x%29=x%2820-pi%2Ax%2F4-x%2F2%29%2B%281%2F8%29pi%2Ax%5E2
(unfinished...).
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Answer by ikleyn(52781) 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 = 40  ====>  y = 20+-+x%2F2+-+%28pi%2Ax%29%2F4.


The area A = xy + %281%2F2%29%2Api%2A%28x%2F2%29%5E2 = x%2A%2820-x%2F2+-+%28pi%2Ax%29%2F4%29 + %28pi%2F2%29%2A%28x%2F2%29%5E2 = 20x - x%5E2%2F2 - %28pi%2F4%29%2Ax%5E2 + %28pi%2F8%29%2Ax%5E2 = -x%5E2%2F2 + 20x - %28pi%2F8%29%2Ax%5E2



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


-x+%2B+20+-+%28pi%2F4%29%2Ax = 0,   or   x%2A%281%2Bpi%2F4%29 = 20  ====> x = 20%2F%281%2Bpi%2F4%29 = 20%2F%281+%2B+%283.14%2F4%29%29 = 11.20 ft.


Answer.  The maximum area is at x = 11.20 ft.

Solved.

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Be aware:   The method on how  @josgaritmetic  is doing it   I S   I N C O R R E C T.