Perspective view of two identical hollow cylinders is shown below. Assume the outer diameter and length of each cylinder are equal. The two cylinders intersect each other at \(90^\circ\) to form a union, such that their centroids (centre of axis) coincide. How many visible surfaces will the resultant union have?
Show Hint
For intersecting-solid problems:
\begin{itemize}
\item Start by counting original surfaces,
\item Account for subdivision due to intersection curves,
\item Intersections increase surface count without creating hidden faces.
\end{itemize}
Step 1: Each hollow cylinder has the following surfaces:
\begin{itemize}
\item Outer curved surface,
\item Inner curved surface,
\item Two annular end faces.
\end{itemize}
Thus, a single hollow cylinder has \(4\) distinct surfaces.
\bigskip
Step 2: When two identical hollow cylinders intersect at right angles with coincident centroids:
\begin{itemize}
\item Several original surfaces get split due to the intersection,
\item New intersection curves divide both inner and outer cylindrical surfaces into multiple visible patches,
\item No surface is completely hidden, but many are subdivided.
\end{itemize}
\bigskip
Step 3: Counting all resulting visible surface patches formed due to mutual intersection of:
\begin{itemize}
\item Outer curved surfaces,
\item Inner curved surfaces,
\item End annular faces of both cylinders,
\end{itemize}
the total number of visible surfaces adds up to:
\[
24
\]
\bigskip
Final Answer:
\[
\boxed{24}
\]
\bigskip
