In ceramic materials composed of ions the mutual coordination of cations by anions and vice versa is determined by their relative ionic radii, r$\_($c$)$ and r$\_($a$),$ respectively. Compute the minimum and maximum values for the ratio $($r$\_($c$))/($r$\_($a$))$ that holds for a $4-$fold $($tetrahedral$)$ coordination. Show clearly $($full derivation$)$ how you obtain your answer.