$400kg$ of spherical dried food particles with a volume of surface mean diameter of $2\mathrm{.}2mm$ are crushed in a hammer mill in $30$min. the power required is $58kW\mathrm{.}80\%$ of the feed passes through $0\mathrm{.}180mm$ clear opening screen. After crushing operation, food powders are separated according to their sizes by screen analysis for screen clear openings of $0\mathrm{.}242,0\mathrm{.}212,0\mathrm{.}180,0\mathrm{.}150,0\mathrm{.}132,0\mathrm{.}100$ and $0\mathrm{.}06mm,$ the mass fractions retained on each screen are $0,0\mathrm{.}00338,0\mathrm{.}0509,0\mathrm{.}1153,0\mathrm{.}1209,0\mathrm{.}6730$ and $0\mathrm{.}0061\%$ respectively. The volume surface mean diameter of crushed product is found as $0\mathrm{.}1275mm.$

a$)$ Prepare the cumulative screen analysis table.

b$)$ Calculate the work index for Bond's law.

c$)$ What will be the power required crushing the same feed to a volume surface mean diameter of $0\mathrm{.}05mm?$