A silo (right circular cylinder) with diameter $10$ meters stands in a field. Alice is located $11$ meters west and $10$ meters south of the center of the silo. Bob is located $11$ meters east and $g>0$ meters south of the center of the silo. The line of sight between Alice and Bob is tangent to the silo. What is the value of $g$?

1. Put the center of the silo at the origin (radius $5$), Alice at $(-11,-10)$ and Bob at $(11,-g)$.
2. The line through Alice with slope $m$ is $mx-y+(11m-10)=0$. Tangency means its distance from the origin is $5$:
   $$\begin{gathered}(11m-10)^2=25(m^2+1)\ \Longrightarrow\ 96m^2-220m+75=0\\ \Longrightarrow\ m=\frac{220\pm140}{192},\end{gathered}$$
   so $m=\frac5{12}$ or $m=\frac{15}8$.
3. Bob is on the line: $-g=-10+22m$, so $g=10-22m$. For $m=\frac{15}8$ this is negative, so $m=\frac5{12}$ and $g=10-\frac{55}6=\boxed{\dfrac56}$.