Areteem: II geometry 9.21 ,9.27, and 9.30

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II geometry 9.21 ,9.27, and 9.30

These questions are ambiguous. I am confused about what they are talking about. For 9.21, how is the sphere placed? is it tangent to the base or is it close to the vertex of the cone? For 9.27, does removing opposite tetahedra mean removing the all other tetahedra and making a regular tetahedron? And what exactly is S? is it a solid, face or a point? For 9.30, we only talked about part b, when the circle is exactly inscribed in a frustrum, how can I find where is the intersection of the circle? Does the theorem apply even if the circle does not inscribe perfectly?

Here's some hints:

For 9.21: I replied recently to a question about this problem here: https://classes.areteem.org/mod/forum/discuss.php?d=1072 The diagram I gave there is a good starting place.

For 9.27, we are just removing the two tetrahedra that are opposite along a face diagonal. For example, if we have a cube ABCD-EFGH (ABCD is the base, and EFGH is above ABCD so that AE, BF, etc. are edges), we are removing tetrahedron A-BDE and G-CFH. One height should be fairly easy. For the other try to work indirectly by subtracting off the removed portions.

For 9.30, we are still talking about the same situation as in class, where the sphere is inscribed inside the frustrum and is tangent to the top, bottom, and the sides (the side portion is what creates the circle the problem is asking about). Hence, you can use the result we proved in class.

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