3.1 Problems related to area
Example $5 \triangle A B C$ has three sides $a, b, c$ satisfying $1 \leqslant a \leqslant 3 \leqslant b \leqslant 5 \leqslant c \leqslant 7$.
Find the perimeter of $\triangle A B C$ when its area is maximized.
Solution: Note that
$$
S_{\triangle A B C}=\frac{1}{2} a b \sin C \leqslant \frac{1}{2} a b \leqslant \frac{1}{2} \times 3 \times 5,
$$

equality holds if and only if $a=3, b=5, \angle C=90^{\circ}$. At this time, $c=\sqrt{a^{2}+b^{2}}=\sqrt{34}$ satisfies $5 \leqslant c \leqslant 7$.

Therefore, when $a=3, b=5, c=\sqrt{34}$, the area of $\triangle A B C$ is maximized. At this time, its perimeter is $8+\sqrt{34}$.