The angle that corresponds to a complete revolution around the lake is [tex]2 \pi[/tex].
Taking the direction of the first person as positive direction, his angular position around the lake is
[tex]\theta_1 (t) = \omega_1 t[/tex]
where [tex]\omega_1 = 1.80 \cdot 10^{-3} rad/s[/tex] is the angular speed of the first person.
The second person is going in the opposite direction, so we can write his angular position around the lake as
[tex]\theta_2 (t) = 2 \pi - \omega_2 t[/tex]
where [tex]\omega_2 = 3.70 \cdot 10^{-3}rad/s[/tex] is his angular speed, and [tex]2 \pi[/tex] is the angle that corresponds to one complete revolution around the lake.
The two people meet when their angular position is the same:
[tex]\theta_1 (t) = \theta _2 (t)[/tex]
[tex]\omega_1 t = 2 \pi - \omega_2 t[/tex]
from which we find the time t after which they meet again:
[tex]t= \frac{2\pi}{\omega_1 + \omega_2} = \frac{2 \pi}{1.80 \cdot 10^{-3} rad/s + 3.7 \cdot 10^{-3} rad/s}=1142 s [/tex]
