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Question
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.
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Solution
We have,
the edge of the cubical piece, a = 21 cm and
the radius of the hemisphere, `r = a/2=21/2`
The surface area of the remaining piece = TSA of cube + CSA of hemisphere -Area of
= 6a2 + 2πr2 - πr2
= 6a2 + πr2
`= 6 xx 21xx21xx22/7xx21/5xx21/2`
`= 21xx21(6 + 22/(7+4))`
`= 21xx21(6+11/14)`
`=21xx21((84+11)/14)`
`= 21xx3(95/2)`
= 2992.5 cm2
Also,
Volume of the remaining piece = volume of the cube - volume of the hemisphere
`= a^3 - 2/3pir^3`
`= 21xx21xx21 - 2/3xx22/7xx(21/2)xx(21/2)xx(21/2)`
`= 21xx21xx21xx(1-2/3xx22/7xx1/2xx1/2xx1/2)`
`= 21xx21xx21(1/1 - 11/42)`
`=21xx21xx21xx(42-11)/42`
`= 21xx21xx(31/2)`
= 6835.5 cm3
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