$10.$ A flat polystyrene$-$air ion chamber with $1-$mm plate separation is positioned with the inner surface of its front wall at a depth of $2\mathrm{.}0$ g$/$cm$^2$ in a polystyrene phantom. A beam of $15-$MeV electrons incident on the phantom makes a charge reading of $6\mathrm{.}4210$ C$,$ where p $=765$ torr, T $=25\mathrm{.}2$C$,$ and the ion collection efficiency is $0\mathrm{.}973.$

$($a$)$ What is the absorbed dose in the cavity air, known to have a mass of $4\mathrm{.}3110^4$ g$?(5$ pts$)$

$($b$)$ What is the absorbed dose in the polystyrene at $2-$g$/$cm$^2$ depth? $(5$ pts$)$