$($a$)$ A liquid flows at a rate of $30k\frac{g}{s}$ into an empty tank of $2m$ in internal diameter. The filling process occurs at a constant temperature and pressure such that the liquid density is $980k\frac{g}{m^3}.$ How long does it take $($in min$)$ to fill the tank to a liquid height of $4m?$

$($b$)$ After reaching a liquid height of $4m,$ a valve at the bottom of the tank is opened. The liquid starts to flow out of the tank through an opening with a diameter of $8cm$ and at a velocity that is proportional to the liquid height, given by $v=\sqrt[\phantom{2}]{2gh},$ where $g$ is the gravitational constant and $h$ is the liquid height. All this time, fresh liquid continues to flow into the tank at the rate of $30k\frac{g}{s}.$

$($i$)$ Show that the liquid level starts to go down when the valve is opened.

$($ii$)$ Calculate the time $($in min$)$ for the liquid height to reduce by half.

$($iii$)$ Will the liquid height reach a steady state value? If so$,$ what is this height? If not, why not?