00:02
A normal window has a shape of a rectangle surmounted by a semi -circle.
00:07
We want to find the dimensions of a normal window of perimeter 25 feet that will admit the greatest possible amount of light.
00:19
So the greatest possible amount of light corresponds to the maximum area of the normal window.
00:28
So we get to find an expression of the area and use the perimeter in some way.
00:35
So the geometrical situation we have here is the following.
00:42
We have a rectangle as the base of the window.
00:47
Let's say it has width w and height h.
00:52
And on top of that, we have a semi -circle.
00:55
Let's do it a little bit better.
01:02
It's a semi -circle.
01:04
And because we have this length here is w again, we can say that the radius, of the circumference correspond to semicircle.
01:20
This radius here, suppose is the center, has the relationship with the width, that the radius is half the width of the rectangle.
01:33
So that's a fundamental part of the construction of the window and the perimeter corresponds to this side, this height of the rectangle, this width, this height and the semi -circference.
01:55
This side here is not included in the perimeter because it's inside the window.
02:04
So we can say that the perimeter of the window is 2 times h plus with one time only plus the length of the semi -circumference.
02:21
The length of the whole circumference is 2 pi r and so the lens of the semi circumference is pi r and now we replace r by w half so we get 2h plus w plus pi times w half that is perimeter is 2h plus w x xx1 plus pf which in fact is 2h plus w plus 2h plus plus w times 2 plus pi over 2.
03:19
And we know the perimeter is 25 feet, so for that we suppose all length are measured in feet.
03:29
So 25 is 2h plus w times 2 pi how 2 plus pi over 2.
03:43
That's a fundamental equation.
03:49
And let's call that asterisk.
03:55
Okay.
03:57
So we write at this moment, we write the area of the window.
04:11
And that corresponds to the sum of the area of the rectangle plus the area of the semicircle.
04:19
It's called that area a.
04:21
And that's equal to the area of the rectangle is w times h with times height plus half of the area of the circle.
04:32
The area of the circle is by r square so half of that is the area of the semicircle replacing r by w half here we get w h plus by half times w half square that is w h plus w square times pi over 8 so the area is w h plus w square times by 8th that's the equation of the area of the window split here area of the norman window and that's the expression or that quantity the area of the window is the quantity we want to maximize okay so in this formula we have two variables, w and h, but we can reduce the formula to one variable only by using this expression of d 'b of h in terms of w that we can get by solving for h this equation asterisk here.
06:35
So from asterisk we get h equal kind of be a fraction.
06:46
So is 25 minus w times 2 pa 2 plus pi over 2 25 minus w half times 2 plus pi over 2 so let me arrange this a bit okay me verify 25 minus w 2 plus pi over 2 over 2 because we have here 2h okay and we can say this is the same as sorry there is a w half is not there and currently let's see that let me verify that here we have w times 2 plus pi over 2 w times 2 plus pi over 2 so there is no 2 here for w so this is equal to 25 over 2 plus sorry minus w half times 2 plus pi over 2 that's h and that expression of a we of h we put it inside the formula of the area above so we can say that a is w times here and 25 half minus w half times 2 plus pi over 2 okay plus w square times by 8th so a of w that is a now depends only on w is 25 half w minus w square over two times two plus five over two plus w plus w square times five eighth now we have w over two common factor between these two terms so the area in terms of w is 25 half w minus and we take on factor with a negative sign also w square over 2 that is negative all w square over 2 are common factor times so here we get 2 plus pi over 2 okay minus because we get a common factor negative here and we must have a positive so inside parenthesis we get negative and when we get this two cofactor we get a 4 here okay because when we distribute this again, we get w square pi over 8.
10:43
Well, here, when we distribute this with the first factor, we get negative w square over 2 times 2 plus pi over 2, which is what we have here.
10:53
So this is the expression...