Determining the inner diameter of a shell of a sphere in water?

Question

A spherical shell of copper with an outer diameter of 15.5 cm floats on fresh water with half its volume above the water's surface. Determine the inner diameter of the shell.

____ cm

Answer 1

So the weight of half of the volume of the sphere is equal to the weight of an equivalent volume of water.

density of water at 20C = 0.998 g/cm³ = 998 kg/m³

density copper 8960 kg/m³ = 8.96 g/cm³

The volume of a half sphere is  of water is

V = ½ * ⁴/₃πr³ = 2/3 π (15.5/2)³

V = 975 cm³

975 cm³ x 0.998 g/cm³ = 973 g or 0.973 kg

The weight of that water is 0.973 kg x 9.8 = 9.53 N which is the upwards force on the sphere

The mass of the half sphere is that same 0.973 kg, (973g) and part is copper, part is vacuum.

Volume of hollow half sphere is:

V = (1/2)(4/3)π(7.75)³ - (1/2)(4/3)π(7.75-d)³

V = 975 - 2.094(7.75-d)³ cm³

Mass of that is 8.96 g/cm³ x 975 - 2.094(7.75-d)³ cm³

M =  8736 - 18.76(7.75-d)³

973 = 8736 - 18.76(7.75-d)³

combine and solve for d

I'm out of time.

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