$(25$ points$)$ Let's now consider a biological treatment of the Sucrase. A pilot$-$scale treatment system is built to test for BOD removal efficiency. The sywem is tested with a concentration of $50m\frac{g}{L}$ of Sucrose. Samples from the influent and effluent of the treatment system are collected and analyzed for BODs. All BOD lests are conducted with a total volume of $300$ mL $,$

The samples are diluted with seeded difution water that has a small amount of BOD ilself. To find the BODs of the dilution witer, an experiment containing only the dilution water was performed. The initial DO of the dilution wioner is $9\mathrm{.}5$ might. If the final DO of the dilution water is $8\mathrm{.}0m\frac{g}{?}.$ after the $5-$day BOD test, what is the BODs of the dilution water?

ii$.$ When testing the strength of the wastewarer, $10$ mL of the waste solution is used. The initial DO of the mixture is $9\mathrm{.}5m\frac{g}{L}.$ If the final DO of the mixture after $5$ days is $3\mathrm{.}25m\frac{g}{L},$ what is the $BO{D}_s$ of the wastewater?

If the pifor treatment system has a substrate removal efficiency of $57\%$ and treats a flow of $0\mathrm{.}05\frac{m^3}{m}in,$ how much $O_2($in kg $)$ would be required for $12$ hours of operation?