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Problem 70 Hard Difficulty

Water in a bowl A hemispherical bowl of radius 8 inches is filled to a depth of $h$ inches, where $0 \leq h \leq 8 .$ Find the volume of water in the bowl as a function of $h$. (Check the special cases $h=0 \text { and } h=8 .)$

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Okay, The goal here and this problem is in this problem 70 goaless. Ah, establish many, Let's say the formula to describe for the calculate the volume off water in the heinous. Critical about Okay, I'm hysterical about right. Ah, as a function off as a function of height. No, that's zero. And ah, at any given height right there. Let's height. That's the height. What is the volume? Hey, that's the question. So we have We have done a lot of figures, problems and in, ah, calculation of volumes of solids and things and one of those secure Remember Waas producing a hemisphere by a rotation off Ah, off polynomial about the access or one of the access so you could think of what he could. But he could be held X y access. We can have maybe a public that and then looted about the x axis and get a boat. Or we could do they could know I haven't talked like this and then rotated about the why access and generate about the all these scenarios. So this looks like this, but except that you simply notice the bottom of the ball is not a religion and That's really what I want. But get get the bottom of the bowl at the origin. What we should do is, uh, might we might need to get a little bit credo and, uh, put some little bit of talking about this and tell Ozols Maskell so out us. How do we do this? How we can really probable backwards. That's the best way. So you with all that the xxx X and y axis. What they answer. If we went toe, you let a ball at the base of the origin of the bottom are the corresponding religion. We know that we can do this. We can imagine a semi circle like that on arc 1/4 I should say, 1/4 off a circle in the X y plane. And then if reiterated this about why but the white Edixon we could get we could generate the the hemispherical all of interest. So all they need is really a co one segment of the scope off that semicircle and our agent around why? And calculate the volume. And so if that's what we need, let's see what ever given we are given that Ah, the really a solid the radius off this him as his goodness. Eight into civil. You This. Okay, So if that zits if this is the bottle betweens, that would be eight. The lead corniness. Sorry. Zero comma. Okay, what is do we have? We have Ah, let's see this limit here. Um, that's it. Thats 00 Everything's a circle that this is the thing. That's the radius between Is this point here? It's eight. Comma zero. Okay, so we have the limits. We have We're gonna will calculate. Let's see, that could be a radius. Okay. And then the alligators are why, and we'll get this. So what are we What do we need? We need this guy here when we don't know this. What is this look like? In other words, Orders ffx like right on a simple white. Or do we need therefore right. But remember, in all of problems, if we are going noted that on why the radius would be in this direction and so the integration would have proceeded have opposite along the way. But this is anyone we need. You need effort by. Yeah. So it couldn't be What? Well, new mess, Some point of view up. Okay? No. And we would have, you know, the limits. Some zero toe ate their parliament. So that's taking, you know, I could let's go ahead and start writing that the formula for the volume and rotation about the Y axis this goes has C to D. This is a disc problem, not awash. Apartment has only one car. That's just by times, um, the radius squared, right? So and we have decided it's going to be It's a function of white. So this is gonna be p o lie, Quinn. The original waters feel what? The same for that you will recognize that this is people used to serve the problem for circle. Actually know that if we completed this, this are completely there'll be a circle. Yeah. So this is really part of part of the equation of a circle, except that the center is not bait is located at 08 So we should Let's you have to walk secure and use that question. The general form the general equation off a circle or her circle. You may have to do some references, but the recorders x minus h the whole square less. Why minus k well, where equals are square. Where h k is the center is the coordinator of the coordinates of the center all of course, radius. But looking at this graph, we just plug will be 1208 to be the center, which means it is zero and K is eight and are also is a You know this, but we can rely This equation of the circle s X minus zero. The whole square loves why minus K's eight. The whole square equals eight square with just 64. Okay, remember, we're looking for in terms off what you owe me an X, in fact, don't like because X is gonna be the radius at any point. So that's my door X simply nordics. Oh, x choir less y minus eight. No one's quality puts it before or X squared equals 64 minus y minus eight. The homes Quick, this is an important relation. Okay, remember, excess the radius and we have x squared. And that's okay because why? Because they were gonna have a square it anyways. But we can leave it as exquisite. Be of why is the same as X? Your boy is the radius which is the same musics existence. This so we just know set up all right off the expression for the volume. Using this, see is zero and D is the upper limit. And why in the skaters were eight So they're taking proceeds considered eight with a volume no one knew and they did arrest by, you know, the eight oh 60 or minus y minus eight. The holds that's off. That's really it was quite see what? So that's that would be the case if we to calculate the volume or the hemisphere from all the way the complete hemisphere. But what we need to finalists find out is what is the volume off water that Elsie Hemisphere toe some arbitrary height H R eight per something less than make Okay, something like like there. That's right. That's stuck with hyper. That's some height, which very calculated the radius. Oh, that's the work in question. Any I bitch was a This is there. It doesn't matter. This is a little phone. If we set up, this doesn't change its wise A very general home. So all we needed changes Is the value off 8 to 8 And that baby No, they would. We will be calculating for the volume of water that is thrilled. More. Write it. The holiest blue because blue waters, you had the idea. Okay, so that's a real volume or some height. A little. So And my the suspects were I our agree height? Well, that was by, but it's not each four y minus eight. Where do you like? Okay, miss, If they don't read, Brian tickles. That's by times zero bridge 64 y minus the integral off. Wyman's ate the whole square list by a huge dreaded by big. They not plug in the A plug in h and M pregnant. Zero and expand. This will be by finds, um, 64. It minus. It's minus eight minus eight. Her Betsy K minus zero minus when you plug in 00 minus minus eight. Cube divided by three. Wait system five times before. Itch minus. It's minus eight. The whole cube by three minus eight. Cuba. This alone this was amigos gonna be this because negative eight Cube, this is is a negative fight well made over this positive, but India's negative order to it, but it means negative. Okay, so that's really the the formula for you're wrong. You mean guns off. Ah, height of any given height. Okay. Yeah. So we don't see this the way to simplify this, But maybe there is, But this is pretty good. Now we can We can confirm we need to impact. That's a good idea. To a disappointing confirm that this makes sense. For instance, if h 10 but the most obvious, they use a nice bodies of H 10 which means you don't feel any water. The volume required the fell. Then there's no one should. This thing just completely vanishes. The volume off water should be. We know intuitively should be zero. Well, does it does it does this volume ah equals zero at X equals zero, but it just plug it in and see what happens at it that at H equals zero. Let's see, like until you get pH equals five times 64 times 00 minus 01 is eight and mine is eight. Cuba's negative by 12 native at night it was will become positive. Pribyl. I fell over three and then minus vital over three just and indeed it Venice's. That's good. Check and now the next. That's not over. We need toe. Confirm what happens if we fill fill the world completely if you filled it. We know that now we have completely filled a volume off hemisphere. That's what we need. The volume of a hand off the hemispheres work. This should ideas too, so we should substitute X equals eight. So at H equals eight, we would have evaluated, be it and then compare if it doesn't match. We used to be conducive formula for the volume off a hemisphere was there. This is eight and confirm that they are identical. They're the same. The let's plug in its equal to eight into this cargo into this former here. And we see that a vertical eight we which equals by times 64 times eight minus eight minus eight. You glad? Likely. That's easy. Minus Phi 12/3. Okay? No, just which is by times 64 times eight is 512 miners. This guy, eight minutes 80 disappears. We have minus mile trail over three. Well, this is then I'm here to by three. By times 5 12 That's what we got using our formula is your knowledge off rotations? Ah Fox two to produce solid bullets. But that's our value. Now we need to compare Check Jacqui the the known for Melo for William off a and he's here. Well, most in all the picture you'll find a volume of the steel and so wonderful hemisphere is just half of that. But we know the volume off sphere off radios are is oh, but by our cube. So while you more Amy's fear Yes, off of that actually to over three by r cubed So what is eight in our case at R equals eight for Article eight It's too over the buy signs it killed just over the by times by 12. That's what the formless is and wanted. We love it when we get is your mother big nation to what required I did or matter is good. No matter was good the beans are monitored which waas Really the real goal was to do this find a volume, uh, at any arbitrary high volume of border quite a film the hemisphere toe up to any