Example $2$: A $5-$mL sample of sewage is diluted to $300$ mL in a standard BOD bottle $.$ The initial DO in the bottle is $9\mathrm{.}2$ mg $/$ L$,$ and after $5$ days of incubation the DO is found to be $4\mathrm{.}7$ mg $/$L $.$ Determine the $5-$ day and ultimate BOD values for the sewage $,$ assuming that the reaction rate constant $($k $)$ is $0\mathrm{.}14/$d $.$