Mikulás is examining the sky. The next day, on his name day, he wants to travel as far as possible to bring gifts to the children. At midnight, it finally starts to snow. Mikulás is a true expert in snowfall. He immediately sees that this is the kind of snowfall that lasts continuously for 24 hours. In the first 16 hours of the snowfall, the sleigh can accelerate faster and faster. (The speed can be increased uniformly and continuously.) At the beginning of the snowfall, the sleigh cannot be moved at all, but by the end of the 16th hour, it can fly like a storm. However, after this, the thickening snow becomes an increasingly greater obstacle, and the achieved speed decreases uniformly over 8 hours to zero. Mikulás does not want to tire the reindeer pulling the sleigh for more than 8 hours. When should Mikulás set out if he wants to travel the longest possible distance?
Solution. We plotted the speed that Mikulás can achieve on a speed-time graph; the slope is 0 at the 0 time point, at the $M=16$ time point he can travel at $m$ speed, and the speed at the 24th hour is again 0. If the sleigh starts at the $S$ time point (at $(16-x$ hours) and, taking advantage of the maximum 8 hours of travel time, arrives at the $T$ time point (at $24-x$ hours), then the distance traveled by the sleigh is exactly the area of the pentagon $A S T B C$, which is the sum of the areas of the (possibly degenerate) trapezoids $A S M C$ and $C M T B$. According to the theorem of parallel chords,

$$
A S: C M=(16-x): 16
$$

thus

$$
A S=m\left(1-\frac{x}{16}\right)
$$

Similarly, $B T: C M=x: 8$, so

$$
B T=\frac{m x}{8}
$$

Therefore,

$$
\begin{aligned}
t_{A S M C}+t_{C M T B} & =\frac{x}{2} m\left(2-\frac{x}{16}\right)+\frac{8-x}{2} m\left(1+\frac{x}{8}\right)= \\
& =\frac{3 m}{32}\left(-\left(x-\frac{16}{3}\right)^{2}+\frac{256}{9}\right)+4 m
\end{aligned}
$$

it is the largest when $x=\frac{16}{3}$; therefore, Mikulás sets off at 10:40 AM.

![](https://cdn.mathpix.com/cropped/2024_05_02_ef588503d0de1b314d89g-1.jpg?height=364&width=588&top_left_y=1016&top_left_x=758)