Approximately $15\%$ of the particles in a shipment of silver$-$bearing ore are judged to be argentite, $)=(7\mathrm{.}3gcm-3,87\%Ag$; the remainder are siliceous $)=(2\mathrm{.}6(g)c{m}^{-3}$ and contain essentially no silver.

$($a$)$ Calculate the number of particles that should be taken for the gross sample if the relative standard deviation due to sampling is to be $1\%$ or less.

$($b$)$ Estimate the mass of the gross sample, assuming that the particles are spherical and have an average diameter of $4\mathrm{.}0mm.$

$($c$)$ The sample taken for analysis is to weigh $0\mathrm{.}600g$ and contain the same number of particles as the gross sample. To what diameter must the particles be ground to satisfy these criteria?