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A cylinder is circumscribed along a hemisphere and a cone is inscribed in the cylinder.
The vertex of the cone is in the center of the base of the cylinder. Prove that

vol. of the cylinder / 3 = vol. of the hemisphere / 2 + vol. of the cone / 1

Solution: Let r be the radius of the hemisphere. Then heights of the cone and cylinder are both r.

1/3 (vol. of the cylinder) = 1/3 (Π r2 r) = 1/3 Π r3

1/2 (vol of hemisphere) = 1/2 (2/3 Π r3) = 1/3 Π r3

vol of the cone = 1/3 Π r2 r = 1/3 Π r3

Therefore, vol of cylinder/3 = vol of hemisphere/2 = vol of cone/1

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