Gravel is being dumped from a conveyor belt at a rate of $10$ cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is $23$ feet high? Recall that the volume of a right circular cone with height $h$ and radius of the base $r$ is given by $V= \frac{1}{3}\pi r^2h$.

When the pile is $23$ feet high, its height is increasing at feet per minute.