Question 8
Torus $T$ is the surface produced by revolving a circle with radius $3$ around an axis in the plane of the circle that is a distance $6$ from the center of the circle (so like a donut).
Let $S$ be a sphere with a radius $11.$ When $T$ rests on the inside of $S,$ it is internally tangent to $S$ along a circle with radius $r_i,$ and when $T$ rests on the outside of $S,$ it is externally tangent to $S$ along a circle with radius $r_o.$ The difference $r_i - r_o$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
Let $S$ be a sphere with a radius $11.$ When $T$ rests on the inside of $S,$ it is internally tangent to $S$ along a circle with radius $r_i,$ and when $T$ rests on the outside of $S,$ it is externally tangent to $S$ along a circle with radius $r_o.$ The difference $r_i - r_o$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$