In this video I find the largest sphere that passes through a given point while every point inside is also inside a larger sphere. Although the problem states every point inside the sphere satisfies an inequality, simply completing the squares multiple times reveals that the inequality is just in the form of a large sphere. Using the radius of the larger sphere, its center, and a unit vector from our point to the center of the big sphere, we can obtain the formula for our inside sphere. I also graph our spheres with the amazing GeoGebra 3D graphing calculator: https://www.geogebra.org/3d/dtdpukqs
Timestamps:
- Problem 6: Largest sphere in a sphere: 0:00
- Solution: Complete the squares first: 0:40
- Plug all the squares back into the equation: 6:00
- Obtain an equation of a sphere: 8:08
- The distance from our point Q to the center of the big sphere P: 10:28
- Visually showing the sphere within a sphere: 13:13
- Desired radius of our smaller sphere: 17:25
- Obtaining the center of smaller sphere using unit vectors: 18:51
- Equation of our smaller sphere: 25:15
- Graphing with GeoGebra: 26:28
Video notes and playlists:
- HIVE notes: https://peakd.com/hive-128780/@mes/jlgigkdl
- Full video and playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0G_7Jd6ZORHFQMk8jbwmkos
- Vectors and the Geometry of Space playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
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