A submarine's dive is modeled by a quadratic function $S(t)$, where $S(t)$ is the depth in meters and $t$ is the time in minutes. The submarine reaches its maximum depth of $100$ meters $5$ minutes after starting its dive and returns to the surface after $10$ minutes. Based on this function, what was the estimated depth, to the nearest meter, of the submarine $2$ minutes after it began its dive?