A Norman window is a rectan-gle with a semicircle on top. Suppose that the perim

Question

A Norman window is a rectan-gle with a semicircle on top. Suppose that the perimeter of a particular Norman window is to be 24 feet. What should its dimensions be in order to maximize the area of the window and, therefore, allow in as much light as possible?

Now assume that the semicircle is stained glass which transmits only half the amount of light as clear glass

Explanation / Answer

perimeter=(2r+2x)+ pi r =24 area=0.5 pi r^2 +2rx =0.5pi r^2 +r(25-pi r-2r) =(0.5pi-pi-2)r^2 +25r=0 derivative=0 ==>r=25/(4+pi) ==>r=3.5feet-->radius of semicircle. x=3 feet (height of rectangle) now when semi circle is stained glass light transmitted is proportional to0.5*0.5 pi r^2 +2rx =(0.25pi-pi-2)r^2 +25r=0 derivative=0 ==>r=25/(4+0.5pi) ==>r=4.487feet-->radius of semicircle. x=0.463 feet (height of rectangle)

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