You are tasked with designing a rectangular open channel to transport a flow rate of $5$m$^3/$s with the minimum cost of construction. The channel slope is $0\mathrm{.}001,$ and the roughness coefficient is $0\mathrm{.}03.$ Determine:

$1.$ depth of the channel $=1\mathrm{.}64$ m

$2.$ hydraulic radius $=0\mathrm{.}82$ m

$3.$ velocity $=0\mathrm{.}92$

A rectangular barge with dimensions of base width $1\mathrm{.}5$m$,$ height of $2$m and length of $8$m is floating in water. The barge has a uniform density of $800$ kg$/$m$^3.$ Compute:

$4.$ draft of the barge $=1\mathrm{.}60$

$5.$ the metacentric height if the center of gravity is $1$m above sea level $=-1\mathrm{.}68$

$6.$ the restoring or upsetting moment when the angle of tilt is $15$ degrees $=102$