13.051. Two cylinders roll down an inclined board 6 m long, one of which has a circumference of 3 dm, and the other 2 dm. Can the circumferences of both cylinders be increased by the same amount so that on the same path one of them makes 3 more revolutions than the other?
Solution.

Assume that the lengths of the circumferences can be increased by $x$ dm. Then the first cylinder will make $\frac{60}{3+x}$ revolutions, and the second will make $-\frac{60}{2+x}$. According to the condition, $\frac{60}{3+x}+3=\frac{60}{2+x}$, from which we get $x^{2}+5 x-14=0$. Solving this equation, we find that the lengths of the circumferences can be increased by $x=2$ dm.

Answer: can be increased by 2 dm.