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Derive an expression for the buoyant force on a spherical balloon, submerged in water, as a function of the depth below the surface, the volume of

Derive an expression for the buoyant force on a spherical balloon, submerged in water, as a function of the depth below the surface, the volume of the balloon at the surface, the pressure at the surface, and the density of the water.  At what depth is the buoyant force half the surface value?
I know that the Buoyant force is equal to rho_water * g * V and that the pressure at the surface is P=P_0  + (rho * g * h).  I'm not sure how to tie it together.

This question was asked on Apr 26, 2010.

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