Consider a cylindrical tank of water with a level indicator on it. The tank has water flowing into (F;(t) inft/min) and out of (F,(t) in ft/min) it. You may assume

that the water flowrate out of the tank is proportional to the height of water in the tank with proportionality constant 1/R. The cross-sectional area of the cylinder is Ac and the density of the water is p. Perform an unsteady-state mass balance on the tank to determine the height for a given F;(t). Do not solve the ODE. Show that it fits the following equation: R A_{c} \frac{d H}{d t}+H=R F_{i} Invert the following functions: \text { a. } \frac{s+6}{s\left(s^{2}+5 s+6\right)} \text { b. } \frac{s+2}{s^{2}+8 s+19} c \frac{2 s+1}{s\left(s^{2}+6 s+9\right)} Take the Laplace Transform of the following ODE to get X(s). Do NOT invert X(s). \frac{d^{3} x}{d t^{3}}+3 \frac{d^{2} x}{d t^{2}}-2 \frac{d x}{d t}+x=1+3 t^{3} \quad x(0)=1 \quad x^{\prime}(0)=0 \quad x^{\prime \prime}(0)=-1 Factor the following polynomial that has three real roots: \text { e. } s^{3}+2 s^{2}-s-2