A box$-$shaped vessel is $8$ m long, $5$ m wide and $3$ m high and it is floating in seawater $(\backslash (\backslash$rho$\_\{\backslash$text $\{$sw $\}\}=\backslash )\backslash (1025\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash )).$ It has a mass of $\backslash (80,000\backslash$mathrm$\{$~kg$\}\backslash )$ and its centre of gravity is located $1\mathrm{.}8$ m from the bottom of the vessel.

$($a$)$ What is the distance from the waterline to the bottom of the vessel? $(4$ marks$)$

$($b$)$ Is the vessel stable when floating in seawater? $(7$ marks$)$

$($c$)$ Cargo is loaded onto the deck of the vessel. The cargo has a mass of $20,000$ kg and the cargo's centre of gravity is located $3\mathrm{.}5$ m from the bottom of the vessel. What the position of the overall centre of gravity of the vessel and its cargo? $(\backslash (\backslash$mathbf$\{3\}\backslash )$ marks$)$

$($d$)$ Is the vessel still stable after this cargo has been loaded? $(6$ marks$)$