sketch-the-indicated-curves-and-surfaces-sketch-the-curve-in-space-defined-by-the-intersection-of--2

Question

Sketch the indicated curves and surfaces.
Sketch the curve in space defined by the intersection of the surfaces $x^{2}+(z-1)^{2}=1$ and $z=4-x^{2}-y^{2}$

Vikash Ranjan · Accepted Answer

In the problem we have the equation X squared plus David -1 Whole Squad Equal one. So this is the arranged equation for this. Now the other equation is get equals four minus X, Squire minus white squad. So we have this as equation one and this has equation two. Now using the situation, we have drawn the car in two dimensional figure, so this is a truly diagram for these equations and this is our excesses, this is a Y axis and this is our taxes, so this is the answer to the problem.

Patrick Vaughn · Answer

and this problem we want to sketch a region and closed by the quadratic surfaces. Z equals X squared plus y squared and Z equals four minus X squared minus wise words. So we have to provide Lloyds. And for this problem, we&#x27;re gonna need to remember, um, some of the characteristics of a privilege and also how we find a curve of intersection. So in order to find the curve of intersection, we&#x27;re going to try to eliminate some of the variables. And in here and one of the things we can dio as we can say, row is equal to row Squared is equal to X squared plus y squared because you&#x27;re kind of gonna be entangled. Otherwise we&#x27;re not gonna be able to have three independent equations. So when we try to cancel out row, we&#x27;re gonna be adding these two equations so we add them, we&#x27;re gonna get to Z is equal to four Z equals two. So already we&#x27;re looking at those equals two plane, which is pretty good. Next, what? We can dio as since we know what&#x27;s in this plane, we could just plug this value into our first equation. So we get to is equal Thio X squared plus y squared or what we could do is we could try subtracting these equations and we get the same thing. So if we subtracted them, we would get, um, zero is equal to negative. Four plus two x squared plus y squared before equals two x squared plus y squared. We get, um, two is equal to X squared plus y squared. Okay, so here we are. So what curve is this? So this is gonna be a circle with radius equal to square to to and with the center of 00 on and to. Okay, so then so now I want to graph the region enclosed. So here we have a crab Lloyd that opens up on the positives, the access. And here we have a problem oId that opens negatively on this, actually, but has a vertex at C equals four. Okay, so let&#x27;s go ahead and do a rough sketch. So here&#x27;s my axes. So, uh, here, we&#x27;re gonna have it opening up with the Vertex of 000 the origin, and then up here, we&#x27;re gonna have it opening down instead. And we&#x27;re gonna have an intersection at Z equals two in the form of a circle. So we get this kind of almost egg shaped, but with, um, with a rim, that&#x27;s kind of sharp, Right? And let&#x27;s go ahead and lots of points. So up here we have 004 and then down here, we have Ah, 000 And then over here, we have, Ah, 0 to 2. And then back there, we would have. Sorry, that would be negative. 20 to and over here, we got 202 and then back at the end of the circle, we have, Ah, zero negative. Two and two. All right, that completes the problem.

Patrick Vaughn · Answer

this problem were given to quadratic surfaces surfaces and we want Thio Sketch the region enclosed by them and describe the curve of intersection. Our services are two x squared plus two y squared plus Z squared equals three and Z equals X squared, plus wise word. And so we want to describe these. So we&#x27;re gonna need to recall, um, some of the properties for a lip sides and Prabal oId because so here we have a no upside here. We have a problem. I&#x27;d and we&#x27;re also gonna need to recall how we find a curve of intersection. So first, we&#x27;re gonna try to find the curve you intersection, because it might make crapping easier. So if we take our second equation and substituted into our first because we can rewrite our first equation. So here&#x27;s one Here&#x27;s to. So by one, we have to x squared plus y squared plus Z squared is equal to three. That&#x27;s two Z plus Z squared minus three is equal to zero. So let&#x27;s reverse boil this. So we get Z plus three times Z minus one is equal zero. So we got Z is equal to negative three or one so it&#x27;s kind of weird because we got to solutions. So let&#x27;s look at this. Z equals three. And let&#x27;s also look at the curve Z equals X squared plus y squared. And since, um X and why are really numbers? Um negative three equals X squared. Plus y squared is a contradiction. Therefore, because we can&#x27;t have a negative value is equal to some of to positive values. Doesn&#x27;t make any sense. Eso Therefore, we&#x27;re gonna throw out. We were just gonna go ahead and throw out this three value. We don&#x27;t need that. So we have Z looking at the planes equals one, and so either we could plug it into our first equation. But the second equation looks pretty easy since we just have one equals X squared plus y squared. So that&#x27;s a a circle with radius equal toe one, and the center is going to be at the point 00 And always disease is equal to 1001 Okay, so was the shape gonna look like? So we have this Prabal oId that opens from the origin up. And then here we have this ellipse oId that kind of lives somewhere in this Prabal oId and the only place where the Intersect is going to be at a circle. So we&#x27;re trying to get a rough idea of the sketch. So we have the privilege, opens up from the origin, and but the Ellipse oId lives somewhere around it, right? And then we get this intersection that&#x27;s a circle on top. So then our region contained in between is gonna be where the where these surfaces overlap pretty much where what&#x27;s both contained in them. So that&#x27;s gonna be inside this Prabal oId and then but cut off by the Ellipse on top, we&#x27;re gonna get kind of like the cut off tip of a banana sort of look. All right, so let&#x27;s go ahead and plot that. So let&#x27;s draw our axes first. And so we don&#x27;t really need to worry about the bottom part of our. So let&#x27;s start with the circle. So we have a circle at. So here&#x27;s one and has a radius of one. And it&#x27;s parallel to the X Y plane I can draw is a little bitter, like hard. There we go. That looks a little better now, all right? And so we have our Ah, lips. OId are sorry are Paraiba. Lloyd is coming up to this and then on top, we have the shell, as you will from this Ellipse oId Yeah. All right. So we want to know We want to draw some points here. So, first of all, what&#x27;s our Vertex? Our vertex is at the origin, so we have 000 And this point is on that circle. So it&#x27;s gonna be 011 and then back here we have negative 10 one. And then up here, we have zero negative one one. And over here, we&#x27;re gonna have zero. Sorry. That should be 101 and then last. We want to find this point. So what&#x27;s that? So we had our lips, which was two x squared plus y squared plus z squared equals three. So, if, uh, this is on the Z axis, then my ex wise or zero, So I get c squared equals serious, So it&#x27;s gonna be easy equal square to three, because it&#x27;s positive. So this is gonna be the point 00 square to three. All right, that completes our problem.

Carlos Pinilla · Answer

So you have this shape. These, uh, question si is equal to x squared plus four y squared. So you want this kitsch that looks like, you know, if your our axes why eggs z were Z equals you, you have, ah, exit square plus more white squares Zeros on wall. That he&#x27;s on is the point. But if you let her see the world so se c equals one. But I want oneness with X squared. It was four. Why squared loss of the season? No upside from since you blew girl to be sent something like these. Well, on a leaps were the plane there See, was one until as you live your C to go, uh, even off the year leaves is so, uh, well, I&#x27;m the since x squared plus four y squared. He&#x27;s not very positive. See, also the positive. So it was like a burnable, Lloyd. So these that has, like, what can a leaps for c constant? Um uh, well, the way decedent lease has to be like, what could wreak is to say I really like Well, that&#x27;s golf over the plane. Uh, physical zero. So, over that lame, we should have that there&#x27;s a why. Well, seaplane, The curve I see for why squared. That looks so something like that. So it is. Ah, like the shape on duh trust bears cold cuts or C constant should be and sex.

Carlos Pinilla · Answer

So you have this absolute for squared miss four y squared waas c squared off the 16 so you can give my life for, like, the Marine get here. So these are four. You know, Where is she Wanted X squared was Why square These will see or is evil, though. Square jail. See, we&#x27;re do squared. Um, so this is entirely I&#x27;m, uh, to return in the and points all the Arabs all day the boundaries eso we consider Sequels You. So that is it playing? We&#x27;re gonna have a circle overuse to, So they should be to go there, do there, And we have the circle on the X Y plane. Well, cereal not should be a circle, and they know we can see there. Well, so that has to people to No true square asked people toe See, we will do squared. Uh, so, uh, si que go to she can go to see for. So see here would go up to four because we&#x27;re running. She was four for X and y equals you. So he&#x27;s, uh he&#x27;s, like, elongated along the see axes like these Where that went appears four now there is minus four. She was like these, Like we said, They said so

Linda Hand · Answer

Okay, first let&#x27;s draw these things separately and then we&#x27;ll put them together and make one picture here. First of all, Z equals the square root of X squared plus y squared. Look for a minute. Let y equal zero. Then you get Z equals the absolute value of x squared. Okay? So in the Xz plane, which is this one we get this line to Z equals x. Z equals X. When x is one, Z is one. So it looks like this except positive Z. Only because it says positive the square root. Okay, so it&#x27;s 45° in the XZ plane. And then when you let um X equals zero, you get Z equals the square root of y squared, or Z equals Y. Which is also a 45° angle. Uh that&#x27;s a terrible picture of me drag getting here. Yeah. All right, here&#x27;s my coordinate system. So the first one is 45° into the first quadrant. Okay, right here and then the 2nd 1 45 degrees this way. So right here. So what this is a picture of is a cone that opens up on the positive Z axis because this is positive the square root and it&#x27;s 45 degrees up from um the xy plane. So 45 degrees here, 45 degrees here. 45 degrees here. Alright, next x squared plus y squared equals one. Well, in two dimensions, that&#x27;s a circle whose radius is 00 and I mean center 00 and its radius is one. So it&#x27;s the circle right here. That&#x27;s the circle like thing. Right here, one more time there it goes. Okay, put them like it&#x27;s a circle. Okay? And it doesn&#x27;t say anything about z, Z can be anything. Nazi is nothing Z is anything. So, what this is is a cylinder. Okay, So it looks like that and Z equals wanted. Is these between one and 2? What? Okay, z equals one. That&#x27;s the xy plane, Z equals two. That&#x27;s the plane parallel to the xy plane. But two units up. All right. So, we got a cylinder. We got uh cone and then we got this cut off front so that it&#x27;s above the xy plane and then cut off at two. All right, So, let&#x27;s see if we can get that. Okay, I think would be useful to draw the cylinder first. It only needs to go up and then inside the cylinder we have this cone And then we&#x27;re gonna cut it off at um Z equals two. And when z equals two, let&#x27;s see where we are on the cone. two equals x squared plus two equals the square root of x squared plus y squared or two for equals X squared plus y squared. Yeah. Okay, so let me think about this minute, let me okay, what I really want to do is start my picture over. Now, I&#x27;ve decided to draw the cone first so they can get a bit drawn better. Okay, It starts here at 00. When it gets to Z equals one, It&#x27;s a circle of Radius one. When it gets to Z equals two. Now it&#x27;s a circle of radius force, so it&#x27;s way big out here. Okay? So there&#x27;s the cone and then the cylinder it&#x27;s cutting right through here. Okay? There it is. Okay, so in the picture We have the cone up to where it gets to one. Z equals one. And then after that we have the cylinder until it gets too too. Okay so it sort of looks like a pencil or pin like that Where this is equal zero, This is equals one and this is equals two. And it&#x27;s cut off here because the cylinder is smaller than the cone is once you get above one. Okay.

Problem 46 Easy Difficulty

Sketch the indicated curves and surfaces.
Sketch the curve in space defined by the intersection of the surfaces $x^{2}+(z-1)^{2}=1$ and $z=4-x^{2}-y^{2}$

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