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
Answers
Answer:
Given,
Length of edge of the cube a=21 cm
diameter of hemisphere d=a=21 cm
hence, radius of the hemisphere r=
2
d
=
2
21
=10.5 cm
Surface area of cube=6a
2
Curved surface area of hemisphere =2πr
2
Area of base of hemisphere=πr
2
Total surface area of remaining block = surface area of cube + surface area of hemisphere - area of base of hemisphere
=6a
2
+2πr
2
−πr
2
=6a
2
+πr
2
=(6×(21)
2
+
7
22
×(10.5)
2
) cm
2
=(6×441+
7
22
×110.25) cm
2
=(2646+346.5) cm
2
Total surface area of remaining block=2992.5 cm
2
Volume of the cube=a
3
Volume of hemisphere=
3
2
πr
3
Hence,
Volume of remaining block = volume of cube - volume of hemisphere
=a
3
−
3
2
πr
3
=(21
3
−
3
2
×
7
22
×(10.5)
3
) cm
3
=(9261−
3
2
×
7
22
×1157.625) cm
3
=(9261−2425.5) cm
3
Volume of the remaining block=6835.5 cm
Step-by-step explanation:
Length of cubical piece of wood = a = 21 cm
Volume of cubical piece of wood = a3
= 213
= 9261 cm3
Surface area of cubical piece of wood = 6a2
= 6 × 21 × 21 cm2
= 2646 cm2
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
∴ Volume of remaining solid is 6835.5 cm3
∴ Surface area remaining piece of solid is 2992.5 cm2