The quadratic function $f$ models the depth, in meters, below the surface of the water of a submarine $t$ minutes after it began its descent. The function estimates that the submarine reached its maximum depth of $200$ meters $5$ minutes after it began its descent and then returned to the surface of the water $10$ minutes after it began its descent. Based on the function, what was the estimated depth, to the nearest meter, of the submarine $8$ minutes after it began its descent?