Q$.1($a$)$ A soil sample with total volume of $8{10}^{-3}{m}^3$ is collected in a construction site for laboratory testing. The total mass of the soil sample is $15\mathrm{.}45kg.$ When dried out in an oven, the total mass of the sample is reduced to $13\mathrm{.}55kg.$ The specific gravity of the soil is $2\mathrm{.}71.$ Determine the following properties of the soil sample:

$($i$)$ moisture content $($w$)$;

$($ii$)$ bulk density of soil $({}_b)$;

$($iii$)$ dry density of soil $(($:${}_{\text{d}\text{r}\text{y}}\}$;

$($iv$)$ density of soil solid, $({}_s)$;

$($v$)$ volumes of air $(V_a),$ water $(V_w)$ and solid grain $(V_s)$;

$($vi$)$ void ratio $($e$)$;

$($vii$)$ porosity $($n$)$;

$($viii$)$ degree of saturation $(S_r)$;

$($ix$)$ saturated density $(($:${}_{\text{s}\text{a}\text{t}}\}$;

$($x$)$ submerged density $(($:${}_{\text{s}\text{u}\text{b}}\}$; and

$($xi$)$ additional mass of water, in $kg,$ to be added to the soil to reach to $100\%$ saturation.

$($b$)$ An engineer compacted the soil sample obtained in Q$\mathrm{.}1($a$)$ and found that the total volume of the compacted soil sample is reduced to $7\mathrm{.}21{10}^{-3}{m}^3.$ Determine the following soil properties of the compacted soil sample:

$($i$)$ void ratio $($e$)$;

$($ii$)$ dry density $({}_{\text{d}\text{r}\text{y}})$; and

$($iii$)$ saturated density $({}_{\text{s}\text{a}\text{t}}).$