SOLUTION: I am stuck with where to begin... I think I need the perimeter of rectangle, area of rectangle and half circumference of circle...
A norman window is a rectangle with a semicircle
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Question 761289: I am stuck with where to begin... I think I need the perimeter of rectangle, area of rectangle and half circumference of circle...
A norman window is a rectangle with a semicircle on top. Design a window requiring 24 ft of trim on outer edges. What dimensions will allow the maximum amount of light to enter a house?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
First thing is to make the rather reasonable assumption that the maximum amount of light would be admitted by the maximum area window.
The perimeter of the overall window is 
Where 
is the height of the rectangular part of the window and 
is the radius of the semi-circular piece.
From this relationship we can determine that r)
The area of the rectangular portion of the window is then r^2)
The area of the semi-circular portion of the window is 
Add 'em up: \ =\ 24r\ -\ \left(2\ +\ \pi\right)r^2\ +\ \frac{\pi}{2}r^2)
Take the first derivative: \ =\ 24\ -\ \left(4\ +\ 2\pi\right)r\ +\ \pi r\ =\ 24\ -\ (4\ +\ \pi)r)
Set the derivative equal to zero: r\ =\ 0)
So 
Take the second derivative: \ =\ -4\ -\ \pi)
which is negative for all values of . So the function is concave down in the region of the extremum and therefore the value of the function at that point is a maximum.
So the rectangular part of the window must have a width of twice the radius of the semi-circle, or 
, and the height must be \left(\frac{24}{4\ +\ \pi}\right\) =\ \frac{24}{4\ +\ \pi})
I'll leave the verification of that simplification to you.
That's all you need to answer your question.
John
My calculator said it, I believe it, that settles it
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