A large conical sand–slurry separator consists of an inverted cone of height 10 m and base radius 5 m. A mixture of water and sand enters through the top at a volumetric rate of 6 m3/min. Simultaneously, a pipe at the bottom withdraws the fluid at a rate that depends on the instantaneous depth h (in meters) of the slurry according to Qout(h)=0.8h1/2 m3/min.
Let h(t) be the depth of the slurry at time t minutes.
(a) Obtain a formula for the volume V of slurry in terms of h only.
(b) Write a differential equation for dtdh in terms of h by combining the rates of volume change with your result from part (a).
(c) Compute dtdh when the slurry depth is 4 m. Include units.
(d) Determine all positive depths h (if any) at which the slurry depth is not changing. Classify each such depth as an equilibrium level where the slurry surface would rise, fall, or remain stable under a small perturbation. Justify your classification.