3. A three-stage launch vehicle consists of stages in the form of cylinders. All these cylinders are similar to each other. The length of the middle stage is two times less than the sum of the lengths of the first and third stages. In the fueled state, the mass of the middle stage is $13 / 6$ times less than the total mass of the fueled first and third stages. Find the ratio of the lengths of the first and third stages. The masses of the engines, instruments, and the stages' shells can be neglected.
Answer: $\frac{7}{5}$

Solution. The mass of the fueled stage is proportional to the third power of the linear size $l_{2}=\frac{1}{2}\left(l_{1}+l_{2}\right), l_{2}^{3}=\frac{6}{13}\left(l_{1}^{3}+l_{2}^{3}\right)$. Solving this system, we get $\frac{l_{1}}{l_{3}}=\frac{7}{5}$ or $\frac{l_{1}}{l_{3}}=\frac{5}{7}$. From engineering considerations, we choose the first answer, since the first stage is usually larger than the third.