$(8$ points$)$ A BOD test is run using $20$ mL of treated wastewater mixed with $200$ mL of pure water. The initial DO of the mix is $8\mathrm{.}0m\frac{g}{L}.$ After $5$ days, the DO is $5\mathrm{.}0m\frac{g}{L}.$ After a long period of time, the DO is $1\mathrm{.}5m\frac{g}{L},$ and it is no longer decreasing with time. Assume there is no nitrogenous BOD and that all BOD is carbonaceous.

a$.$ What is the ultimate BOD?

b$.$ What is the $5-$day BOD of the wastewater?

c$.$ What would be the remaining BOD after five days have elapsed?

d$.$ Estimate the reaction rate constant, day $($:${?}^{-1}$