3. Find the minimum value of the expression

$$
\sqrt{a^{2}+1}+\sqrt{b^{2}+9}+\sqrt{c^{2}+25}
$$

if it is known that $a+b+c=12$.
Solution 1. Introduce vectors $\boldsymbol{m}(a ; 1), \boldsymbol{n}(b ; 3)$ and $\boldsymbol{p}(c ; 5)$ and set $\boldsymbol{q}=\boldsymbol{n}+\boldsymbol{m}+\boldsymbol{p}$. Since $a+b+c=12$, then $\boldsymbol{q}(12 ; 9)$. Since $|\boldsymbol{q}|=|\boldsymbol{n}+\boldsymbol{m}+\boldsymbol{p}| \leqslant|\boldsymbol{n}|+|\boldsymbol{m}|+|\boldsymbol{p}|$, then

$$
15=\sqrt{12^{2}+9^{2}} \leqslant \sqrt{a^{2}+1}+\sqrt{b^{2}+9}+\sqrt{c^{2}+25}
$$

Equality is achieved if the vectors $\boldsymbol{m}(a ; 1), \boldsymbol{n}(b ; 3)$ and $\boldsymbol{p}(c ; 5)$ are collinear, $\frac{a}{1}=\frac{b}{3}=\frac{c}{5}$, from which it follows that $b=3 a$ and $c=5 a$. Therefore, $a+3 a+5 a=12$, from which $a=\frac{4}{3}, b=4, c=\frac{20}{3}$.

Answer: 15.