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Numerade Educator

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Problem 24 Easy Difficulty

Find an equation of the largest sphere with center $ (5, 4, 9) $ that is contained in the first octant.

Answer

$(x-5)^{2}+(y-4)^{2}+(z-9)^{2}=16$

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Video Transcript

Okay, So here were asked to find an equation of the largest sphere with center five comma four common nine that has continued the first often. And here what is meant by Oct int is equivalent to the first quadrant in the X Y plane. But here we're in X Y Z space. So we're not talking about first quadrants anywhere. We're talking about first ock tent, which is where x, y and z all non negative. So it's like the positive ah, space that we're living in. No, the largest fear containing the first Optimus having radius that's equal to the minimum distance from the center to any of the three coordinate plaints. And that's because the sphere has re contained in the first option. So it can't intersect any of the old coordinate planes or going to the other options. So we have to have the rate is equal a minimum distance from the center to any of the planes. Now, the minimum distance he's gonna be given by an absolute value of the smallest coordinate, right? So the shortest such distance is to the X Z plane, because the Y cornet is smallest, the X Z plane is what we have left, and so we have the shortest distance. It's a distance of four. So now that we're given the center, we can write the equation of the sphere as X minus five squared plus Y minus four squared Posey minus nine. Squared is equal to the the radius, which is the distance to the X Z plane, the shortest distance. So this is equal the radius squared, which is 16 units.