Given
#h->"the depth of the well"=12m#
This will also be the length if the rope
#M->"the mass of bucket+ water in it"=15kg#
#W->"the work done to raise the bucket"#
#"filled with water"=2160J#
Let the mass of the rope be m kg
The total work done against the gravitional pull will be
#W=Mgh+mg*h/2#
Here #mg*h/2# is the work done to raise the center of gravity ( at #h/2# depth) of the rope.
So
#Mgh+mg*h/2=2160#
#=>15*9.8*12+m*9.8*12/2=2160#
#=>m=(2160-15*9.8*12)/(9.8xx6)=6.73kg#
So linear density of the rope
#="mass of the rope (m)"/"length of the rope (h)"#
#=6.73/12~~0.56" kg/m"#