In many locations, such as Lake Washington in Seattle, floating bridges are pref

Question

In many locations, such as Lake Washington in Seattle, floating bridges are preferable to conventional bridges. Such a bridge can be constructed out of concrete pontoons, which are essentially concrete boxes filled with air, Styrofoam, or another extremely low-density material. Suppose a floating bridge pontoon is constructed out of concrete and Styrofoam, which have densities of 2200 kg/m3 and 50.0 kg/m3. What must the volume ratio of concrete to Styrofoam be if the pontoon is to float with 31.8% of its overall volume above water?
V concrete/V styrofoam =

Explanation / Answer

They give you the following densities
density of concrete,c = 2200 kg/m3
density of styrofoam s=50 kg/m3

they don't give you the density of water, but from the description "ie lake washington" we can assume that it is going to be freshwater. Assuming water at 15 deg C, fw= 1000 kg/m3

let volume of concrete be Vc and volume of styrofoam be Vs

Total volume of concrete + styrofoame = (Vc + Vs)

mass of concrete = c.Vc ---------- (eq1)

mass of styrofoam= s.Vs ---------- (eq2)

It tells us that 40.3% or the total volume is above water, which means 59.7% (or 0.597) is below water.

Hence the submerged volume = 0.597(Vc + Vs)

The Mass of the submerged voume = fw 0.597 (Vc + Vs) ---------- (eq3)

We know from Archemedes' Principle that the mass of water displaced by the floating bridge (submerged volume) = mass of the floating bridge.

So (eq1) + (eq2) = (eq3)

c.Vc + c.Vc = fw 0.597 (Vc + Vs)

rearranging, you get

Vc/Vs = (0.597fw - s ) / (c - 0.597fw) = [(0.597)(1000) - (50) ] / [(2200) - (0.597)(1000)] = 0.341 (Ans)

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