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.

0

This is important problem from the cbse board exam point of view because this problem was last asked in 2014 cbse board exam. This problem from RS Aggarwal book page number 289 chapter volume and surface area of solid.

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 πr^{3}

= 2/3 × 22/7 × 10.5 × 10.5 × 10.5 cm^{3}

= 2425.5 cm^{3}

Surface area of hemisphere = 2πr^{2}

= 2 × 22/7 × 10.5 × 10.5 cm^{2}

= 693 cm^{2}

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 = 9261cm^{3} – 2425.5 cm^{3}

= 6835.5 cm^{3}

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 = 6a^{2} – πr^{2} + 2πr^{2}

= (2646 – 22/7 × 10.5^{2} + 693) cm^{2}

= 2992.5 cm^{2}

∴ Volume of remaining solid is 6835.5 cm^{3}

∴ Surface area remaining piece of solid is 2992.5 cm^{2}

Length of cubical piece of wood = a = 21 cm

Volume of cubical piece of wood = a

^{3}= 21

^{3}= 9261 cm

^{3}Surface area of cubical piece of wood = 6a

^{2}= 6 × 21 × 21 cm

^{2}= 2646 cm

^{2}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 πr

^{3}= 2/3 × 22/7 × 10.5 × 10.5 × 10.5 cm

^{3}= 2425.5 cm

^{3}Surface area of hemisphere = 2πr

^{2}= 2 × 22/7 × 10.5 × 10.5 cm

^{2}= 693 cm

^{2}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 = 9261cm

^{3}– 2425.5 cm^{3}= 6835.5 cm

^{3}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 = 6a

^{2}– πr^{2}+ 2πr^{2}= (2646 – 22/7 × 10.5

^{2}+ 693) cm^{2}= 2992.5 cm

^{2}∴ Volume of remaining solid is 6835.5 cm

^{3}∴ Surface area remaining piece of solid is 2992.5 cm

^{2}