1. (2 points) According to the standards of the International Crabbing Federation, the crabbing field consists of two areas, one of which is square, and the second has the same width, with a length of 25 to 30 meters inclusive. All dimensions must be whole numbers of meters, and the total area of the field must be between 250 and 300 square meters (inclusive). Find the largest and smallest possible sizes of the square area.

Answer: Side of the square is 7 or 8. Area is 49 or 64.
Solution:

Let the side of the square be $x$. When the length of the rectangle is minimal and the area of the field is maximal, we find the maximal $x$ and vice versa, when the length of the rectangle is maximal and the area of the field is minimal, we find the minimal $x$.

We solve the equation $x^{2}-30 x=250$, obtaining $x=-15+\sqrt{475}$. This number lies between 6 and 7.

We solve the equation $x^{2}-25 x=300$, obtaining $x=\frac{-25+\sqrt{1825}}{2}$. This number lies between 8 and 9.

We need the larger roots of these equations, as the smaller ones are negative.

From this, we conclude that the integer $x$ can be among the numbers from 7 to 8 inclusive.