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Numerade Educator

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Problem 38 Medium Difficulty

Describe in words the region of $ \mathbb{R}^3 $ represented by the equation(s) or inequality.

$ x^2 + y^2 + z^2 > 2z $

Answer

all the points outside and not on the sphere of radius 1 centered at $(0,0,1)$

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Video Transcript

So the inequality giving Waas X squared plus y squared plus C squared bitter end Susie. Now, if you look at it lives inside of inequality, you get a crew that shot probably looking. That's a sphere. It's some restrictions apply to it. So now the question of a sphere generally is giving us X minus the square plus Y minus B squared plus C minus C square. Peoples are split where the center is giving us give, you said. As this ends up with this, Will you have the experiment is being the Y being you being seen? And the radius of this year being are now in the case where you have your quality, it means that you're describing the surface off this year when it's a great Indian sign, which means that you're describing the region extent off the street. But return and means that extent also this year, but not including the service. If it were a great Adam or you want to the entrance of the region, extend out to destroy you and also include in its size. So in this case you're interested in their region in his regional this year, like the region extend our industry. Now, if you look at the equation the general form of a question of this year and in the inner quality getting you realize that it doesn't really fit like there's some semblance but we have to do some little so a little work to make sure that we get the same for you us is described. So what we do is we bring the two Z to the seams. I will look like things. Together we end up with X squared plus y squared plus C squared minus two is better now extinct by already satisfied therefore the general for the problems. So how do we change Dizzy to Fitz what we want. So one thing we notice Z squared minus two. Izzy is the same as the squared minus Tracy plus one minus one. Now, if you look at Z squared minus two z plus one and rode with the same A Z minus one or square. So in the end, the Z squared minus twos is the same. A C squared minus one. Boss, we're minus one. If you fit this equation back into the inequality, you end up with a square plus y squared plus C squared minus one minus one. All squared, minus one reset. And your group. You just transferring your my next one thing to the right side of you. End up with X squared plus rice with class Z minus one course with grace. A dumb one. So now the center of this year 001 And then I read your seats. One. So it means that the inequality describes the region extend out to this year off. We just want if sent as it was yours.