The logistic equation characterizes cell growth in terms of carrying capacity $(x_{}).$ The specific growth rate is assumed to be proportional to the unused carrying capacity as follows

$=k(1-\frac{x}{x_{}})$

where $k=$ carrying capacity coefficient and $x=$ cell concentration at time $t$

a$.$ Consider cell growth in a batch reactor. Derive an expression for the concentration of cells $($x$)$ as a function of time $(t)$ in a batch reactor if cell growth follows the logistic equation. What is the physical significance of parameters $x_{}$ and $k?$

b$.$ Ethanol fermentation from glucose is accomplished in a batch culture of Saccharomyces cerevisiae, and the following data were obtained.

$\backslash$table$[[\backslash$table$[[$Time$],[($h$)]],\backslash$table$[[$Glucose$],[s(\frac{g}{L})$