A liquid level system in provided in Figure.

$($a$)$ Derive a complete mathematical for the system.

$($b$)$ Derive the transfer function relating the rate of change of flow rate $q_2$ with respect to input

flow rate $q.$

$($c$)$ If the input flow rate $q(t)$ is a constant rate of $q_i$ for $t>0$ and if the first tank is taken away

such that $q(t)$ is fed directly into the second tank, sketch the time response of the resulting

output flow.