Mathematics

Mathematics

Problem

Two solid shapes are constructed using the same base radius rr and same total height HH. Shape A is a cylinder with a hemisphere on top. Shape B is a cylinder with a cone on top. If the height of the cylindrical part in Shape A is twice the radius (hcyl=2rh_{\text{cyl}} = 2r), and the height of the cylindrical part in Shape B is also 2r2r, determine which shape has the greater volume and by how much (in terms of π\pi and r3r^{3}).

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