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Here are some hints for problems 9.27 to 9.30:

9.27: There are two cases: the solid is resting on a face that was originally a face of the cube, or the solid is resting on a face that wasn't part of a face of the cube. In each case we are looking for the distance between the face that is on the ground to the opposite face of the solid. In the second case, consider the line segment that joins the vertices that were used to form the tetrahedra that were removed; the height of the solid resting on one of this equilateral triangle faces will be the length of the intersection of this line segment with the solid.

9.28: Let $P$ and $Q$ be the intersections of $\overline{AF}$ and $\overline{DC}$, and $\overline{BF}$ and $\overline{CE}$. Show that $P$ and $Q$ are the midpoints of their respective segments. Use this and the fact that the the volume of pyramid $ABC-F$ is one third of the volume of the solid to find the volume of $PQC-F$. 

9.29: Let $I$ be the midpoint of $\overline{AD}$. Find the volumes of $ABC-D$, $EIH-D$ and $FAG-EIH$, and use these to find the volumes of the two solids.

9.30: Consider a side view of the frustrum:

So, in this diagram $EP$ is the radius of the circle where the sphere and the side of the frustrum meet.

Let us know if you need additional hints for these problems. Don't forget to let us know what you have tried.