Diameter of the base of a water-filled inverted right circular cone is $26$ cm. A cylindrical pipe, $5$ mm in radius, is attached to the surface of the cone at a point. The perpendicular distance between the point and the base (the top) is $15$ cm. The distance from the edge of the base to the point is $17$ cm, along the surface. If water flows at the rate of $10$ meters per minute through the pipe, how much time would elapse before water stops coming out of the pipe?