$q,$ Consider a vertical cylindrical tank initially operating at a steady$-$state liquid level of $6ft$ with the inflow rate of $10f\frac{t^3}{m}in.$ The tank cross$-$sectional area is $78\mathrm{.}54f{t}^2.$ The outflow rate of the tank is given by $F_{\text{o}\text{u}\text{t}}=\frac{H}{k},$ where $H$ is the height of the liquid in the tank and $k$ is a constant related to pipe and valve resistance. $1.$ Develop the dynamic equations for this system. $2.$ Assuming that initially the system is at steady state and then $F_{in}$ is increased by $20\%$ at time equal to zero. Determine the time to achieve $90\%$ of the new steady$-$state liquid level.