5. An astronomer discovered that the intervals between the appearances of comet $2011 Y$ near planet $12 I V 1961$ are consecutive terms of a decreasing geometric progression. The three most recent intervals (in years) are the roots of the cubic equation $t^{3}-c t^{2}+350 t-1000=0$, where $c-$ is some constant. What will be the duration of the next interval until the appearance of the comet?
5. Writing the factorization of the polynomial: $t^{3}-c t^{2}+350 t-1000=\left(t-t_{1}\right)\left(t-t_{1}\right)\left(t-t_{1}\right)$, where $t_{1}, t_{2}, t_{3}$ are the roots of the polynomial, we get:

$$
\left\{\begin{array}{c}
t_{1}+t_{2}+t_{3}=c \\
t_{1} t_{2}+t_{2} t_{3}+t_{3} t_{1}=350 \\
t_{1} t_{2} t_{3}=1000
\end{array}\right.
$$

Property of a geometric progression: $t_{2}^{2}=t_{1} t_{3}$. Then $t_{2}^{3}=1000, t_{2}=10$.

Substituting $t_{2}$ into the equation, we find $c=35$. Therefore,

$$
\left\{\begin{array}{c}
t_{1}+t_{3}=25 \\
t_{1} t_{3}=100
\end{array}\right.
$$

From this, $t_{1}=20, t_{3}=5$. Thus, the next time interval is 2.5 years. Answer: In 2.5 years.