228. In a regular quadrilateral prism $A B C D A_{1} B_{1} C_{1} D_{1}$, the height is half the length of the side of the base. Find the maximum value of the angle $A_{1} M C_{1}$, where $M$ is a point on the edge $A B$.
228. Let the height of the prism be $1, |A M|=x$. Circumscribe a circle around triangle $A_{1} M C_{1}$. Consider the solid obtained by rotating the arc $A_{1} M C_{1}$ of this circle around the chord $A_{1} C_{1}$. The angle $A_{1} M C_{1}$ will be the largest if the line $A B$ is tangent to the surface of the resulting solid. This happens if the lines $M O$ and $A B$, where $O$ is the center of the circle circumscribed around triangle $A B C$, are perpendicular, meaning the line $M O$ divides $A_{1} C_{1}$ in the ratio $\frac{|A M|}{|M B|}=\frac{x}{2-x}$.

On the other hand, it can be shown that $M O$ divides $A_{1} C_{1}$ in the ratio $\frac{\left|A_{1} M\right| \cos \widehat{A_{1} C_{1} M}}{\left|C_{1} M\right| \cos \overline{C_{1} A_{1} M}}$. Expressing the sides and cosines of the angles $\triangle A_{1} M C_{1}$ in terms of $x$, we obtain the equation

$$
\frac{\left(1+x^{2}\right)(4-x)}{x\left(9-4 x+x^{2}\right)}=\frac{x}{2-x} \Leftrightarrow x^{3}+3 x-4=0
$$

from which $x=1$. The maximum value of the angle $A_{1} M C_{1}$ is $\frac{\pi}{2}$.