Problem $1.$

A rectangular barge $15$ m long and $4$ m wide weights $100$ tons and its center of gravity is $2$ m above the

keel.

$(1)$ Show that when floating upright in seawater it is in unstable equilibrium.

$(2)40$ tons of solid ballast are now placed in the barge at $0\mathrm{.}8$ m above the keel. Calculate the resulting

transverse metacentric height.