$6\mathrm{.}9.$ Pseudomonas $sp.$ has a mass doubling time of $2\mathrm{.}4h$ when grown on acetate. The saturation constant using this substrate is $1\mathrm{.}3\frac{g}{l}($which is unusually high$),$ and cell yield on acetate is $0\mathrm{.}46g$ cell$/$g acetate. If we operate a chemostat on a feed stream containing $38\frac{g}{l}$ acetate, find the following:

a$.$ Cell concentration when the dilution rate is one$-$half of the maximum

b$.$ Substrate concentration when the dilution rate is $0\mathrm{.}8D_{\text{m}\text{a}\text{x}}$

c$.$ Maximum dilution rate

d$.$ Cell productivity at $0\mathrm{.}8D_{\text{m}\text{a}\text{x}}$

$($Courtesy of E$.$ Dunlop from "Collected Coursework Problems in Biochemical Engineering," compiled by H$.$ W$.$ Blanch for $1977$ Am$.$ Soc. Eng. Educ. Summer School. $$ American Society for Engineering Education.$)$