Posted by T.Gracken on November 08, 2002 at 11:56:56:
In Reply to: Re: Please write the WHOLE question as it was presented to you! (n/t) posted by amy on November 08, 2002 at 09:41:37:
: A norman window is shaped like a rectangle surmounted by a semicircle. Suppose the semicircular part is made of colored glass while the rectangular part is made of clear glass. The colored glass transmits only half as much light per square foot as the clear glass.
: If the window is to have a perimeter of 30 feet, what dimensions for the window maximize the light transmitted?
someone else will probably find an easier way. until then, here's one:
let w = width of window (base width and also the diameter of the semi-circle)
let h = height of rectangular portion of window (i.e. straight side part)
then perimeter is side + side + base + half circle circumference
or P = 2h + w + pi*(w/2) ...[remember diameter is w so radius is w/2 and half the circumference is pi*r]
so 30 = 2h + w + pi*(w/2)
solve for h to get h = 15 - (w/2) - pi*w/4
area of the rectangle portion of window is h*w, which is (using substitution) same as [15 - (w/2) - pi*w/4]*w
area of circlular portion (half a circle) is (1/2)pi*(w/2)2 = (pi*w2)/8
let L = the amount of light coming through the window.
then L = amount of light allowed through rectangle part + amount of light allowed through semi-circle part
and since amount allowed through semi-circle is half that of rectangle we have
L = [15 - (w/2) - pi*w/4]*w + (1/2)*(pi*w2)/8
or L = [15 - (w/2) - pi*w/4]*w + (pi*w2)/16
I will let you simplify the right side. But now just maximize L (that is, determine critical values for L, etc., etc., etc.)
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