Water is added at varying rates to a 300-liter holding tank. When a valve in a discharge line is opened, water flows out at a rate proportional to the height and hence to the volume V of water in the tank. The flow of water into the tank is slowly increased and the level rises in consequence, until at a steady input rate of 60.0L/min the level just reaches the top but does not spill over. The input rate is then abruptly decreased to 20.0L/min.

(a) Write the equation that relates the discharge rate, v _out (L/min), to the volume of water in the tank. V (L), and use it to calculate the steady-state volume when the input rate is 20 L/min.

(b) Write a differential balance on the water in the tank for the period from the moment the input rate is decreased (t = 0) to the attainment of steady state (r → ∞), expressing it in the form dV/d: =.... Provide an initial condition.

(c) Without integrating the equation, use it to confirm the steady-state value of V calculated in part (a) and then to predict the shape you would anticipate for a plot of V versus t. Explain your reasoning.

(d) Separate variables arid integrate the balance equation to derive an expression for V (t). Calculate the time in minutes required for the volume to decrease to within 1% of its steady-state value.