$($b$)$ A skydiver having mass $\backslash ($ m $\backslash )$ diving with the standard skydiving posture. The body of the skydiver is flat and horizontal with spread arms and legs. The cross$-$sectional area of the skydiver's body is $\backslash ($ A $\backslash ).$ The density of air is $\backslash (\backslash$rho $\backslash ),$ and the drag coefficient is $\backslash ($ C$\_\{$d$\}\backslash ).$

$($i$)$ Find an expression for the terminal velocity $\backslash ($ v$\_\{$t$\}\backslash )$ of the skydiver.

$(8$ marks$)$

$($ii$)$ If the skydiver deploys a parachute that takes the shape of a hemisphere having drag coefficient $\backslash ($ C$\_\{$p$\}\backslash ),$ find the expression for the parachute's cross$-$sectional area $\backslash ($ A$\_\{$p$\}\backslash )$ if the safe landing velocity is $\backslash ($ v$\_\{$s$\}\backslash ).$ In this calculation, the drag force acting on the skydiver is negligible compared to that on the parachute.

$(7$ marks$)$