3. Four settlements are located at the vertices of a square with a side length of $10 \mathrm{~km}$. Is it possible to connect these settlements with a road network such that the total length of the roads is less than $28 \mathrm{~km}$, and each settlement is connected to every other settlement?
3. If straight roads connect $A C$ and $B D$, as shown in Figure 5, the road network clearly meets the second requirement, however
$$
A C+B D=2 \cdot 10 \cdot \sqrt{2}>28
$$

does not satisfy the first requirement.
Suppose $E$ and $F$ are the midpoints of the opposite sides $B C$ and $A D$ of the square $A B C D$, and points $K$ and $M$ lie on the line segment $E F$ such that $E K=M F$, denoted as $E K=x$. Roads are connected from $B K, C K, A M, D M$ and $K M$, as shown in Figure 6. This road network clearly also meets the second requirement. Let $s(x)$ be the total road length, then
$$
\begin{array}{l}
s(x)=10-2 x+4 \sqrt{x^{2}+25} . \\
s^{\prime}(x)=\frac{4 x}{x^{2}+25}-2 .
\end{array}
$$

When $x=\frac{5}{\sqrt{3}}$, $s^{\prime}(x)=0$.
At the same time, $s(5 / \sqrt{3})=10(\sqrt{3}$
$$
+1)<28 \text{ km. }
$$

Therefore, when $x=\frac{5}{\sqrt{3}}$, the first road network in Figure 6 meets the first requirement.
(Note: There is no need to find the optimal shape, as long as $s(x)<28$. For example, when $\mathrm{EK}=\mathrm{MF}=\frac{10}{3}$, $s(x)=\frac{10}{3}(2 \sqrt{13}+1)<28$, which meets the requirements of the problem.)
Figure 6