(Solved): Mathematical Modelling: Solution of Laplace Equation 1.- The Liquid-Level system shown in Fig. 1 h ...

Mathematical Modelling: Solution of Laplace Equation 1.- The Liquid-Level system shown in Fig. 1 has a cross section area \( A=2\left[\mathrm{~m}^{2}\right] \). Furthermore, the output valve presents a flow resistance \( R=2\left[\min / \mathrm{m}^{2}\right] \). Considering that the tank is empty at \( t \leq 0[\mathrm{~s}] \), and the transfer function that relates the input flow \( q_{i}(t) \) with the liquid level height \( h(t) \) is \[ \frac{H(s)}{Q_{i}(s)}=\frac{R}{R A s+1} \] answer the following: Fig. 1 Liquid-Level system. a) Applying an input step of magnitude 3; Which is the liquid level height at \( 4[\mathrm{~min}] \) ? b) Plot the liquid level height at any given time \( t \) and support your response. David Sotelo, Ph.D. Carlos Sotelo, Ph.D. 2.- Find the Inverse Laplace Transform of the following function: \[ F(s)=\frac{3}{s(s+1)^{3}} \]