A well$-$mixed lake with constant volume $5$x$106$ m$3$ is fed continuously by a pollution free stream with a flow rate of $850,000$ m$3/$day$.$ A factory continuously dumps $215,000$ m$3/$day of water containing a non$-$conservative pollutant with concentration $100$ g$/$m$3$ into the lake. The pollutant has a first order decay coefficient of $0\mathrm{.}02$ day$-1.$

a$)$ Assuming the pollutant is well mixed in the lake, find the steady state concentration of pollutant in the lake.

After the lake reaches steady state with the waste stream from part a$.,$ the factory installs a new treatment system which reduces the waste concentration to only $5$ g$/$m$3($flow remains the same$).$

b$)$ Calculate how long $($after the treatment is installed$)$ it will take for the concentration in the lake to drop below the water quality standard of $2\mathrm{.}0$ g$/$m$3.$