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In this question, a norman window has the shape of a rectangle surmounted by a semicircle.
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We want to find the dimensions of a norman window of perimeter 35 feet that will emit the greatest possible amount of light.
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So let's draw a picture.
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We have a rectangle and it is surmounted by a semicircle.
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So let's say that the radius of this semicircle is r.
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And let's say that this height is 8.
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So now we're talking about the perimeter being 35 feet.
00:43
So what's my perimeter? well, i've got 2r along the base.
00:49
The two sides are each h, so that's plus 2h.
00:55
While the semicircle up top, the circumference of a circle, is 2 pi r.
01:02
So a semicircle is pi r.
01:06
So 2r plus 2h plus pi r, that's equal to 35.
01:15
Now given that constraint, what do i need to do? i am trying to maximize the area of this window.
01:27
So what's the expression for the area of this window? well, the rectangle is 2r h.
01:33
Then i have half a circle.
01:37
I have one half of a pi r squared.
01:40
Now, i'm trying to maximize this area function subject to this constraint.
01:47
I'm going to have to go to my constraint and solve for one of the variables.
01:51
It's much easier to solve for h.
01:54
I'll say, okay, 2h is equal to 35 minus 2r minus pi r.
02:06
So what does that make h itself? it's 35 halves minus r minus pi over 2.
02:15
And so now i'm going to take that expression for h and i'm going to plug it in.
02:23
So what's my area? 2r times h...