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Numerade Educator



Problem 8 Easy Difficulty

Describe and sketch the surface $ \mathbb{R}^3 $ represented by the equation $ x^2 + z^2 = 9 $.


It is a cylinder with radius $3,$ with axis as $y$ -axis

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Video Transcript

So for this problem we're supposed to describe in three dimensions what this equation looks like now I'm going to apologize ahead of time. I'm a math person, not an artist. So I want to do my best with sketch. Um, I'll try to be as exact as I can with my descriptions so that my sketch makes sense of So please bear with me. So we have a three dimensional grid here. We're going to let X y and Z axes bia's. I've shown them here, and we want to see what this equation looks like. X squared plus Z squared equals nine. Now, as you can see, there is no why in this equation. So I'm going to start by graphing this where? Why is zero? I'm just gonna do this on the X Z plane. If I didn't have why here? This is a circle circle with Radius three. Remember, we have We would have one variable square plus the second variable squared equals the radius squared. And this would be centered if I was just in the XY plane. This would be centered at the origin because I'm not adding or subtracting anything to the X or the Z. So again, ignoring that. Why? You can imagine a circle with a center where X and Z meet with a radius of three. Now, why, though, could be anything for this circle I just picked at. Why? Being zero. But I could move this circle up and down my Y axis so I could move it out here. I'm still gonna have a circle of radius three along my XY plane. But I could move it along my y axis wherever I wish, because I could have y be positive. Five. Negative 12. Whatever. I can slide it along my y axis. So when you take a circle and you slide it, you get a cylinder. So this shape is this equation is going to give us a cylinder, um, along the y axis and the cylinder will have a radius of three. So that's my shape in three dimensions.