To determine the change in height of the water in a cylinder, we need to use the formula for the volume of sphere and cylinder.
Volume of sphere is:
`V_s= 4/3piR^3` where R is the radius
Initila volume of cylinder is:
`V_c=Bh_o` where B is the area of the base
`h_o` is the height of the water
When the sphere is dropped into the cylinder, the volume of the cylinder increases. The total volume is:
`V_T= V_s + V_c`
`V_T = 4/3piR^3 + Bh_o`
Note that the total volume is equal to `Bh_f` where `h_f` is the new height of the water. And B, area of the base, remains the same.
`Bh_f = 4/3piR^3 + Bh_o`
`Bh_f - Bh_o = 4/3piR^3`
`B(h_f-h_o) = 4/3 piR^3`
`h_f-h_o = (4/3piR^3)/B`
`Deltah = (4piR^3)/(3B)`
Hence, the height of water displaced by a metal sphere dropped into the cylindepr is `Delta h =(4piR^3)/(3B)` .