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Question
A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.
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Solution
Diameter of hemisphere = Edge of cube = l
Radius of hemisphere = `l/2`
Curved surface area of hemisphere = 2πr2
= `2 xx pi xx l/2 xxl/2 xx (pi l^2)/2`
Base area of the hemisphare = πr2
= `pi (l/2)^2 = (pil^2)/4`
Surface area of the cube = `6 xx l^2 = 6l^2`
Surface area of the remaining solid = [Total surface area of cube + C.S.A. of hemispjere − base area of hemisphere]
= `6l^2 + (pil^2)/2 − (pil^2)/2`
= `(24l^2 + 2pil^2 − pil^2)/4`
= `(24l^2 + pil^2)/4`
= `l^2/4 (24 + pi)` sq. units.
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