A fermentation broth containing microbial cells is filtered through a vacuum filter. The broth is fed to the filter at a rate of $100$ kg$/$h$,$ which contains $4\%($w$/$w$)$ cell solids. In order to increase the performance of the process, filter aids are introduced at a rate of $12$ kg$/$h$.$ The concentration of vitamin in the broth is $0\mathrm{.}09\%$ by weight. Liquid filtrate is collected at a rate of $94$ kg$/$h; the concentration of vitamin in the filtrate is $0\mathrm{.}042\%($w$/$w$).$ Filter cake containing cells and filter aid is removed continuously from the filter cloth.$($a$)$ Whatpercentagewateristhefiltercake?$($b$)$ If the concentration of vitamin dissolved in the liquid within the filter cake is the same as that in the filtrate, how much vitamin is absorbed per kg filter aid?