From a cubical piece of wood of side 21 cm, a hemisphere is carved out in such a way that the diameter of the hemisphere is equal to the side of the cubical piece. Find the surface area and volume of the remaining piece.

Length of cubical piece of wood = a = 21 cm


Volume of cubical piece of wood = a3


= 21 × 21 × 21 cm3


= 9261 cm3


Surface area of cubical piece of wood = 6a2


= 6 × 21 × 21 cm2


= 2646 cm2


Since, a hemisphere is carved out in such a way that the diameter of the hemisphere is equal to the side of the cubical piece.


So, diameter of hemisphere = length of side of the cubical piece


Diameter of hemisphere = 21 cm


Radius of hemisphere = r = Diameter ÷ 2 = 21/2 cm = 10.5 cm


Volume of hemisphere = 2/3 πr3


= 2/3 × 22/7 × 10.5 × 10.5 × 10.5 cm3


= 2425.5 cm3


Surface area of hemisphere = 2πr2


= 2 × 22/7 × 10.5 × 10.5 cm2


= 693 cm2


A hemisphere is carved out from cubical piece of wood


Volume of remaining solid = Volume of cubical piece of wood – Volume of hemisphere


Volume of remaining solid = 9261cm3 – 2425.5 cm3 = 6835.5 cm3


Surface area remaining piece of solid = surface area of cubical piece of wood – Area of circular base of hemisphere + Curved Surface area of hemisphere


Surface area remaining piece of solid = 6a2 – πr2 + 2πr2


= (2646 – 22/7 × 10.52 + 693) cm2


= 2992.5 cm2


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