35. A hemispherical depression is cut out from one face of a cubical wooden block of edge 21 cm. sa
the diameter of the hemisphere is equal to the edge of the cube. Determine the volume and total
area of the remaining block.
Answers
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
3