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Problem 62 Hard Difficulty

A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area $ A $ of the window as a function of the width x of the window.


$15 x-\frac{x^{2}(\pi+4)}{8}$

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Video Transcript

all right. So here we have a Norman window, a rectangle with a semi circle on top, and we want to find the area of the window is a function of the width. So I'm calling that a of X, and so we're gonna work with the perimeter as well as we go, and we have this rectangle and we don't know its height. So let's call that why? And then we have the semicircle and we know it's radius is X over two. So let's figure out an expression for the area and then we'll work on simplifying it. The area is the area of the rectangle x times y plus the area of the semicircle. So you have to get half the area of a circle half of pi r squared and our is X over two. Okay, so there's half a pyre squared the area of a semi circle. Now we have a formula for area. But the problem is, it has why in it. And we only wanted to have X in it as a variable. It's supposed to be a function of one variable, not too. So we need to find a way to get rid of why, and that's where the perimeter comes in. Okay, so the perimeter of this figure would be X plus. Why, plus another Y, plus half the circumference of the circle. So the perimeter is to why, plus X plus. Now, what's the perimeter of a circle? The circumference of a circle circumference of a whole circle is two pi r. So circumference of half circle would be pi. Times are, and what's R R is X over two. So the perimeter is two y plus X plus pi times X over two, and we know that perimeter equals 30. So this is an equation that we can solve for y. And once we know why, we can substitute it back into our area equation. So to solve this for why, let's start by subtracting X and subtracting high over two x from both sides. We have 30 minus X minus pi over two. X equals two. Why now? We need to divide both sides by two. So that would give us 15 minus 1/2 X minus pi over four. X equals. Why Now? Let's see if we can combine these X terms in some way. If we factor X Out of there, we have X times a quantity 1/2 plus pi over four. Maybe we'll deal with that more later. Okay, This is the quantity, for better or worse that we're going to substitute in for why in our area equation. So now we have a of X equals x times that why I value that we just found as lovely as it is. Plus 1/2 times pi times x over two squared. Now we have some simplifying to dio. Okay, so, uh, let's see here. All right. At this point, I'm thinking that it didn't really help made a factor my ex out at this point. So I'm going to multiply back through, and I'm also going to distribute the X. So we have 15 x minus 1/2 X squared minus pi over four times X squared. Plus. Now over here, let's go ahead and simplify that. When we square X over two, we get X squared over four and we already have a 1/2. So we're going to have high over eight times X squared now. Surely we can combine a couple of these terms in some way. We can add negative pi over four to positive pi over eight and that would give us positive. That would give us negative pi over eight. So we have 15 X minus 1/2 X squared minus pi over eight x squared. Okay, if you wanted to leave the area like that, I would be perfectly happy with it. However, I noticed that the answer in the book is slightly different. So let's see what they dio. If we take the last two terms and we factor out X squared and we factor at 18 we end up with a of X equals 15 x minus X squared over eight times four plus pi. And that's pretty much what you'll see when you look up The answer if you have that, if you have access to that. But there are lots of equivalent forms of the answer that are all totally fine.