Question 159, Let $\mathrm{C}$ be a circle in the $\mathrm{xy}$ plane with radius 1 and center at $O(0,0,0)$, and let a point $\mathrm{P}(3,4,8)$ be in space. If a sphere is completely contained within the oblique cone with base $\mathrm{C}$ and vertex $\mathrm{P}$, find the maximum volume of this sphere.
Question 159, Answer: Let the perpendicular foot of point $\mathrm{P}$ on the $\mathrm{xy}$ plane be $\mathrm{Q}$, then the section of plane $P Q Q$ with the oblique pyramid is a $\triangle$ $P A B$, as shown in the figure. When the great circle of the sphere is the incircle of $\triangle \mathrm{PAB}$, the volume of the sphere is maximized. At this time, the radius of the sphere $r=$ $\frac{2 S_{\triangle A B C}}{P A+P B+A B}=\frac{2 \times 8}{10+2+5 \sqrt{5}}=3-\sqrt{5}$. Therefore, the maximum volume is $\frac{4(3-\sqrt{5})^{3} \pi}{3}$.