232. The bases of a truncated pyramid are regular triangles $ABC$ and $A_{1}B_{1}C_{1}$ with sides 3 and 2 respectively. The segment connecting vertex $C_{1}$ to the center $O$ of the base $ABC$ is perpendicular to the bases; $\left|C_{1}O\right|=3$. A plane is drawn through vertex $B$ and points $M$ and $N$ - the midpoints of edges $A_{1}B_{1}$ and $B_{1}C_{1}$. Consider cylinders located inside the polyhedron $ABC A_{1} M N C_{1}$, with bases in the face $A_{1} M N C_{1}$. Find: a) the maximum value of the volume of such cylinders with a given height $h$; b) the maximum value of the volume among all considered cylinders.
232. Let's consider the section of the polyhedron $A B C A_{1} \pi \uparrow N C_{1}$ by a plane at a distance $h$ from the plane $A_{1} B_{1} C_{1}$, and project the resulting section onto the plane $A_{1} B_{1} C_{1}$ (Fig. 48). In the figure, the projection of this section is denoted by dashed lines. It is clear that the circle of the cylinder's base must lie inside the trapezoid $K L N C_{1} (K, L-$ are the points of intersection of $A_{1} C_{1}$ and $M N$ with the projection of this section). If $h=3$, the section plane coincides with the plane $A B C$, and the points $K$ and $L$ coincide with the midpoints of the sides $B_{1} C_{1}$ and $A_{1} C_{1}$. If $h=\frac{3}{2}$, this radius equals the radius of the circle inscribed in an equilateral triangle with side $|K C|=2-\frac{h}{3}$, i.e., it equals $\left(2-\frac{h}{3}\right) \frac{\sqrt{\overline{3}}}{6}$.

Answer: a) if $0<h \leqslant \frac{3}{2}, V=\frac{3}{16} \pi h$; if $\frac{3}{2}<h \leqslant 3$, $V=\frac{\pi}{12} h\left(2-\frac{h}{3}\right)^{2} ;$

b) the maximum volume will be at $h=2, V=\frac{8 \pi}{27}$.