Problem

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Problem 84 Hard Difficulty

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). The perimeter of the window is 16 feet.

(a) Write the area $ A $ of the window as a function of $ x $.
(b) What dimensions will produce a window of maximum area?

Answer

a. $x\left(8-\frac{x}{2}\left(1+\frac{\pi}{2}\right)\right)+\frac{\pi x^{2}}{8}$
b. $8-\frac{2}{3 \pi / 8+1}(1+\pi / 2)$

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Video Transcript

here. We're given that the perimeter is sixteen. But we could also find the perimeter just by looking at the figure you have X and then you have two of these wise and then you also have this part of the circumference of the circle. You know, the circumference of a circle is two pi r. But since we have half a circle a semi circle well, define that by two. In our case, the radius of the circle. It's just X over too. And so now we can go ahead and well, that'LL answer part a sow writing the perimeters of function of X. So excuse me writing the perimeter and then to find the area. Well, you just take the area of the rectangle, which is X Y, plus the area of the circle. We know the area of entire circles Pi r squared over, dividing by two for half a circle. And then now if you take this equation up here, let's solve that for why? And then plug that back into this. Why over here here And by doing that, we obtain the formula for the area but on ly in terms of the variable x the little answer party for us. And so now the next step is to figure out what X value will give us the dimension of the window with Max area. So really, what we should be looking for is the Vertex of this problem down here. So let's go on to the next page to simplify this. So this is our area and unless you simplify this a little bit so I'll pull out of X squared, it's and then perhaps we can just clean this up a little bit. So this becomes and so now we're dealing with a quadratic that more or less look something like this and the Vertex is always given by this point here. So we should be taking negative B over two A. So the be is the number in front of the X. In our case, that's eight. So don't forget the negative here, some minus a over two times a day. This could be simplified a little bit, cancel the minuses and then cancel the over to us for and this could be simplified a little bit, but there's no need for this. So this is one of the dimensions. This's the X value and to find the Y value we can go ahead and use our formula from the previous page. The formula for why So let's recall what that was. So first, let me just record this X value. Okay, so going on to the next page, remember that we had So this gave us and then you divide it by two. So the last step here isn't just plug in our dimension for eggs Don't forget the one in the two on the bottom here I'Ll just write That is the times one half this can be simplified Let's not simplify this too much here This may be simplified but it'LL still give you the y value of the dimension So there was two dimensions x and y So here is the y values and we just found the X value from the previous page. I'll just remind you of that and there it is. So these two values are the X and Y values that give the maximum area of the window

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