I have a question that I'm struggling with:A raft with a length of 8.0m, a a width of 1.6m, a thickness of 0.11m, and a density of 690.0 kg/m^3 is placed in a river. How far does it sink below it?

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Use Archimede's buoyancy principle.  The weight of the water pushed out of the way by the raft equals the buoyant force which must equal the weight of the raft if it floats.  The raft volume is 1.408 m^3  from (lwh) The raft weight equals dgV where d = density so its 9530.6 N The volume of water displaced by the raft is (8)(1.6)H = 12.8H  where H is the number you want to find. To find the weight of this water you need to know the density of water.  That is 1 g / cc or 1000 kg / m^3 Using the same formula dgV to find the weight of the water we get (1000kg/m^3)(9.81m/s^2)(12.8H) = 125,568H Set this equal to the raft weight 125, 568H = 9530.6 N H = 0.0759 m   The raft floats and is about 3.41 cm above the water.