4.28. One of the sides of a pentagon has a length of 30 cm. The lengths of the other sides are expressed as integers and form an arithmetic progression with a difference of 2 cm, and the length of the smallest side does not exceed 7 cm. Find the lengths of the sides of all pentagons that meet these conditions.
4.28. Let the side $A B$ of the pentagon $A B C D E$ be 30 cm (Fig. 4.25). Further, let $B C=x$; then $C D=x+2$, $D E=x+4$, $E A=x+6$, where $x \in Z, x \leq 7$. For the pentagon to be formed, the inequality $x+x+2+x+4+x+6>30$ must be satisfied, or $4 x>18$, from which $x>4.5$. Thus, from the condition $4.5<x \leq 7$, we find the integer values of the sides: $5,7,9,11,30$; $6,8,10,12,30$; $7,9,11,13,30$.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-586.jpg?height=240&width=571&top_left_y=126&top_left_x=584)

Fig. 4.25

Answer: $5,7,9,11,30$ cm; $6,8,10,12,30$ cm; $7,9,11,13,30$ cm.