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Buoyant force and surface value

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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 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.

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This solution is provided in approximately 158 words. It uses step by step equations to solve for the buoyant force and forces acting on the surface of the water. A diagram is attached in a .jpg file to enhance understanding.

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Because volume is a part of the sphere, where radius makes an angle Q(read as theta) with the vertical, given by

V' = 2*pi*(1 - cos(Q))*R^3/3

If theta < 90 degrees
V' = (2*pi*R^2/3)*(R - ...

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