Assume you are pumping a heavier mud of $12$ ppg at the bottom of the annular space between the drill string and the hole. The total vertical depth is $8,000$ ft and the fracture pressure at the bottom is $4,500$ psi. The heavier mud is displacing the original mud of $9$ ppg$.$ What is the maximum allowable height of the heavier mud in the annulus $($measuredfrom bottom$)$ for not fracturing at bottom?