a$.$ Given a cylindrical tank to be used for irrigation with $1000m^3$ volume $($V$).$ Suppose the tank has to be filled with water from a stream using a pump at a volumetric flow rate, F$,$ of $2\mathrm{.}5L{s}^{-1}.$ Suppose the radius is $10m.$ Suppose at the time of starting to assess the rate of filling the tank is already quarter filled, using the mass balance approach, calculate how long, in days, it will take to fill up the tank. Please make all the necessary assumptions, give a sketch with boundaries, derive the formula from mass balance equations.

$[10$ marks$]$

b$.$ Suppose that, for the tank in $3$a$.$ above, when the water starts spilling $($i$.$e$.,$ when the outflow starts at the same rate as the inflow$),$ a nutrient, phosphate $(P{O}_4),$ is dosed into the pond along with the inflowing water at a concentration of $30mg{L}^{-1}.$

i$)$ Making the sketch, assumptions, mass balances and giving the working equation, if the required level of the phosphate is $3mg{L}^{-1},$ how long will it take to reach the concentration and thus switch to $3mg{L}^{-1}$ phosphate in the inflowing water to maintain the required level? $[7$ marks$]$

ii$)$ What will be the concentration of phosphate $(P{O}_4^{3-})$ after $6$ days of dozing?